You just need to include this in your build.gradle file:

implementation "com.google.android.gms:play-services-location:15.0.1"

or if you're not using latest gradle version:

compile "com.google.android.gms:play-services-location:15.0.1"

Note: It's recommended to use Google Play services version 15.0.1 or higher, which includes bug fixes for this class. More details here.

https://developers.google.com/android/reference/com/google/android/gms/location/FusedLocationProviderClient

You just need to call mFusedLocationClient.removeLocationUpdates(mLocationCallback) in onPause() of your Activity. However, there is a bit more to it than just that.

Use member variables for the FusedLocationProviderClient and LocationRequest in your main activity:

import com.google.android.gms.location.FusedLocationProviderClient;
import com.google.android.gms.location.LocationCallback;
import com.google.android.gms.location.LocationRequest;
import com.google.android.gms.location.LocationResult;

public class MainActivity extends AppCompatActivity
        GoogleApiClient.ConnectionCallbacks,
        GoogleApiClient.OnConnectionFailedListener {

    FusedLocationProviderClient mFusedLocationClient;
    LocationRequest mLocationRequest;

    //..........

Use a member variable for the LocationCallback as well:

LocationCallback mLocationCallback = new LocationCallback(){
    @Override
    public void onLocationResult(LocationResult locationResult) {
        for (Location location : locationResult.getLocations()) {
            Log.i("MainActivity", "Location: " + location.getLatitude() + " " + location.getLongitude());

        }
    };

};

Then, assign mFusedLocationClient in onCreate() :

@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_main);

    mFusedLocationClient = LocationServices.getFusedLocationProviderClient(this);
    requestLocationUpdates();

    //...............
}

Then in onResume(), if theFusedLocationProviderClient is set up, then use it.

@Override
public void onResume() {
    if (mFusedLocationClient != null) {
        requestLocationUpdates();
    }
}

public void requestLocationUpdates() {
    mLocationRequest = new LocationRequest();
    mLocationRequest.setInterval(120000); // two minute interval
    mLocationRequest.setFastestInterval(120000);
    mLocationRequest.setPriority(LocationRequest.PRIORITY_BALANCED_POWER_ACCURACY);
    if (ContextCompat.checkSelfPermission(this,
            Manifest.permission.ACCESS_FINE_LOCATION)
            == PackageManager.PERMISSION_GRANTED) {
        mFusedLocationClient.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper());
    }
}

And finally, in onPause(), call removeLocationUpdates():

@Override
public void onPause() {
    super.onPause();
    if (mFusedLocationClient != null) {
        mFusedLocationClient.removeLocationUpdates(mLocationCallback);
    }
}

This is similar to my other answer here, updated to use the recently introduced FusedLocationProviderClient class.

In order to use a FusedLocationProviderClient in conjunction with a Google Map:

1. Wait until the Google Map is ready

2. Request the Location permission at runtime if needed

3. Request location updates once the permission is granted

4. Update the Google Map once the user's location is obtained

First ensure that you're using at least version 11 of Google Play Services, as older versions don't have the FusedLocationProviderClient class (newer versions will work as well):

dependencies {
      implementation 'com.google.android.gms:play-services-maps:17.0.0'
      implementation 'com.google.android.gms:play-services-location:17.0.0'
    //........
}

Note that the FusedLocationProviderClient is present in version 11.0.2, but due to bugs in the initial implementation, it's recommended that you only use this class on 11.6.0 and later. From the documentation:

Note: It's recommended to use Google Play services version 11.6.0 or higher, which includes bug fixes for this class.

Add the location permissions in the AndroidManifest.xml file, inside the manifest tag and outside of the application tag:

<uses-permission android:name="android.permission.ACCESS_COARSE_LOCATION"/>
<uses-permission android:name="android.permission.ACCESS_FINE_LOCATION"/>

Add the API key to the AndroidManifest.xml inside the application tag:

    <meta-data
            android:name="com.google.android.geo.API_KEY"
            android:value="AIza___YOUR_KEY_HERE______"/>

Kotlin

Here is the full Activity class in Kotlin:

class MapsActivity : AppCompatActivity(), OnMapReadyCallback {

    lateinit var mGoogleMap: GoogleMap
    var mapFrag: SupportMapFragment? = null
    lateinit var mLocationRequest: LocationRequest
    var mLastLocation: Location? = null
    internal var mCurrLocationMarker: Marker? = null
    internal var mFusedLocationClient: FusedLocationProviderClient? = null

    internal var mLocationCallback: LocationCallback = object : LocationCallback() {
        override fun onLocationResult(locationResult: LocationResult) {
            val locationList = locationResult.locations
            if (locationList.isNotEmpty()) {
                //The last location in the list is the newest
                val location = locationList.last()
                Log.i("MapsActivity", "Location: " + location.getLatitude() + " " + location.getLongitude())
                mLastLocation = location
                if (mCurrLocationMarker != null) {
                    mCurrLocationMarker?.remove()
                }

                //Place current location marker
                val latLng = LatLng(location.latitude, location.longitude)
                val markerOptions = MarkerOptions()
                markerOptions.position(latLng)
                markerOptions.title("Current Position")
                markerOptions.icon(BitmapDescriptorFactory.defaultMarker(BitmapDescriptorFactory.HUE_MAGENTA))
                mCurrLocationMarker = mGoogleMap.addMarker(markerOptions)

                //move map camera
                mGoogleMap.moveCamera(CameraUpdateFactory.newLatLngZoom(latLng, 11.0F))
            }
        }
    }

    override fun onCreate(savedInstanceState: Bundle?) {
        super.onCreate(savedInstanceState)
        setContentView(R.layout.activity_maps)

        supportActionBar?.title = "Map Location Activity"

        mFusedLocationClient = LocationServices.getFusedLocationProviderClient(this)

        mapFrag = supportFragmentManager.findFragmentById(R.id.map) as SupportMapFragment?
        mapFrag?.getMapAsync(this)
    }

    public override fun onPause() {
        super.onPause()

        //stop location updates when Activity is no longer active
        mFusedLocationClient?.removeLocationUpdates(mLocationCallback)
    }

    override fun onMapReady(googleMap: GoogleMap) {
        mGoogleMap = googleMap
        mGoogleMap.mapType = GoogleMap.MAP_TYPE_HYBRID

        mLocationRequest = LocationRequest()
        mLocationRequest.interval = 120000 // two minute interval
        mLocationRequest.fastestInterval = 120000
        mLocationRequest.priority = LocationRequest.PRIORITY_BALANCED_POWER_ACCURACY

        if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
            if (ContextCompat.checkSelfPermission(
                    this,
                    Manifest.permission.ACCESS_FINE_LOCATION
                ) == PackageManager.PERMISSION_GRANTED
            ) {
                //Location Permission already granted
                mFusedLocationClient?.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper())
                mGoogleMap.isMyLocationEnabled = true
            } else {
                //Request Location Permission
                checkLocationPermission()
            }
        } else {
            mFusedLocationClient?.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper())
            mGoogleMap.isMyLocationEnabled = true
        }
    }

    private fun checkLocationPermission() {
        if (ActivityCompat.checkSelfPermission(
                this,
                Manifest.permission.ACCESS_FINE_LOCATION
            ) != PackageManager.PERMISSION_GRANTED
        ) {
            // Should we show an explanation?
            if (ActivityCompat.shouldShowRequestPermissionRationale(
                    this,
                    Manifest.permission.ACCESS_FINE_LOCATION
                )
            ) {
                // Show an explanation to the user *asynchronously* -- don't block
                // this thread waiting for the user's response! After the user
                // sees the explanation, try again to request the permission.
                AlertDialog.Builder(this)
                    .setTitle("Location Permission Needed")
                    .setMessage("This app needs the Location permission, please accept to use location functionality")
                    .setPositiveButton(
                        "OK"
                    ) { _, _ ->
                        //Prompt the user once explanation has been shown
                        ActivityCompat.requestPermissions(
                            this@MapsActivity,
                            arrayOf(Manifest.permission.ACCESS_FINE_LOCATION),
                            MY_PERMISSIONS_REQUEST_LOCATION
                        )
                    }
                    .create()
                    .show()


            } else {
                // No explanation needed, we can request the permission.
                ActivityCompat.requestPermissions(
                    this,
                    arrayOf(Manifest.permission.ACCESS_FINE_LOCATION),
                    MY_PERMISSIONS_REQUEST_LOCATION
                )
            }
        }
    }

    override fun onRequestPermissionsResult(
        requestCode: Int,
        permissions: Array<String>, grantResults: IntArray
    ) {
        when (requestCode) {
            MY_PERMISSIONS_REQUEST_LOCATION -> {
                // If request is cancelled, the result arrays are empty.
                if (grantResults.isNotEmpty() && grantResults[0] == PackageManager.PERMISSION_GRANTED) {

                    // permission was granted, yay! Do the
                    // location-related task you need to do.
                    if (ContextCompat.checkSelfPermission(
                            this,
                            Manifest.permission.ACCESS_FINE_LOCATION
                        ) == PackageManager.PERMISSION_GRANTED
                    ) {

                        mFusedLocationClient?.requestLocationUpdates(
                            mLocationRequest,
                            mLocationCallback,
                            Looper.myLooper()
                        )
                        mGoogleMap.setMyLocationEnabled(true)
                    }

                } else {

                    // permission denied, boo! Disable the
                    // functionality that depends on this permission.
                    Toast.makeText(this, "permission denied", Toast.LENGTH_LONG).show()
                }
                return
            }
        }// other 'case' lines to check for other
        // permissions this app might request
    }

    companion object {
        val MY_PERMISSIONS_REQUEST_LOCATION = 99
    }
}

Java

Here is the full Activity class in Java:

public class MapsActivity extends AppCompatActivity
        implements OnMapReadyCallback {

    GoogleMap mGoogleMap;
    SupportMapFragment mapFrag;
    LocationRequest mLocationRequest;
    Location mLastLocation;
    Marker mCurrLocationMarker;
    FusedLocationProviderClient mFusedLocationClient;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_maps);

        getSupportActionBar().setTitle("Map Location Activity");

        mFusedLocationClient = LocationServices.getFusedLocationProviderClient(this);

        mapFrag = (SupportMapFragment) getSupportFragmentManager().findFragmentById(R.id.map);
        mapFrag.getMapAsync(this);
    }

    @Override
    public void onPause() {
        super.onPause();

        //stop location updates when Activity is no longer active
        if (mFusedLocationClient != null) {
            mFusedLocationClient.removeLocationUpdates(mLocationCallback);
        }
    }

    @Override
    public void onMapReady(GoogleMap googleMap) {
        mGoogleMap = googleMap;
        mGoogleMap.setMapType(GoogleMap.MAP_TYPE_HYBRID);

        mLocationRequest = new LocationRequest();
        mLocationRequest.setInterval(120000); // two minute interval
        mLocationRequest.setFastestInterval(120000);
        mLocationRequest.setPriority(LocationRequest.PRIORITY_BALANCED_POWER_ACCURACY);

        if (android.os.Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
            if (ContextCompat.checkSelfPermission(this,
                    Manifest.permission.ACCESS_FINE_LOCATION)
                    == PackageManager.PERMISSION_GRANTED) {
                //Location Permission already granted
                mFusedLocationClient.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper());
                mGoogleMap.setMyLocationEnabled(true);
            } else {
                //Request Location Permission
                checkLocationPermission();
            }
        }
        else {
            mFusedLocationClient.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper());
            mGoogleMap.setMyLocationEnabled(true);
        }
    }

    LocationCallback mLocationCallback = new LocationCallback() {
        @Override
        public void onLocationResult(LocationResult locationResult) {
            List<Location> locationList = locationResult.getLocations();
            if (locationList.size() > 0) {
                //The last location in the list is the newest
                Location location = locationList.get(locationList.size() - 1);
                Log.i("MapsActivity", "Location: " + location.getLatitude() + " " + location.getLongitude());
                mLastLocation = location;
                if (mCurrLocationMarker != null) {
                    mCurrLocationMarker.remove();
                }

                //Place current location marker
                LatLng latLng = new LatLng(location.getLatitude(), location.getLongitude());
                MarkerOptions markerOptions = new MarkerOptions();
                markerOptions.position(latLng);
                markerOptions.title("Current Position");
                markerOptions.icon(BitmapDescriptorFactory.defaultMarker(BitmapDescriptorFactory.HUE_MAGENTA));
                mCurrLocationMarker = mGoogleMap.addMarker(markerOptions);

                //move map camera
                mGoogleMap.moveCamera(CameraUpdateFactory.newLatLngZoom(latLng, 11));
            }
        }
    };

    public static final int MY_PERMISSIONS_REQUEST_LOCATION = 99;
    private void checkLocationPermission() {
        if (ContextCompat.checkSelfPermission(this, Manifest.permission.ACCESS_FINE_LOCATION)
                != PackageManager.PERMISSION_GRANTED) {

            // Should we show an explanation?
            if (ActivityCompat.shouldShowRequestPermissionRationale(this,
                    Manifest.permission.ACCESS_FINE_LOCATION)) {

                // Show an explanation to the user *asynchronously* -- don't block
                // this thread waiting for the user's response! After the user
                // sees the explanation, try again to request the permission.
                new AlertDialog.Builder(this)
                        .setTitle("Location Permission Needed")
                        .setMessage("This app needs the Location permission, please accept to use location functionality")
                        .setPositiveButton("OK", new DialogInterface.OnClickListener() {
                            @Override
                            public void onClick(DialogInterface dialogInterface, int i) {
                                //Prompt the user once explanation has been shown
                                ActivityCompat.requestPermissions(MapsActivity.this,
                                        new String[]{Manifest.permission.ACCESS_FINE_LOCATION},
                                        MY_PERMISSIONS_REQUEST_LOCATION );
                            }
                        })
                        .create()
                        .show();


            } else {
                // No explanation needed, we can request the permission.
                ActivityCompat.requestPermissions(this,
                        new String[]{Manifest.permission.ACCESS_FINE_LOCATION},
                        MY_PERMISSIONS_REQUEST_LOCATION );
            }
        }
    }

    @Override
    public void onRequestPermissionsResult(int requestCode,
                                           String permissions[], int[] grantResults) {
        switch (requestCode) {
            case MY_PERMISSIONS_REQUEST_LOCATION: {
                // If request is cancelled, the result arrays are empty.
                if (grantResults.length > 0
                        && grantResults[0] == PackageManager.PERMISSION_GRANTED) {

                    // permission was granted, yay! Do the
                    // location-related task you need to do.
                    if (ContextCompat.checkSelfPermission(this,
                            Manifest.permission.ACCESS_FINE_LOCATION)
                            == PackageManager.PERMISSION_GRANTED) {

                        mFusedLocationClient.requestLocationUpdates(mLocationRequest, mLocationCallback, Looper.myLooper());
                        mGoogleMap.setMyLocationEnabled(true);
                    }

                } else {

                    // permission denied, boo! Disable the
                    // functionality that depends on this permission.
                    Toast.makeText(this, "permission denied", Toast.LENGTH_LONG).show();
                }
                return;
            }

            // other 'case' lines to check for other
            // permissions this app might request
        }
    }
}

activity_maps.xml:

<?xml version="1.0" encoding="utf-8"?>
<androidx.constraintlayout.widget.ConstraintLayout
        xmlns:android="http://schemas.android.com/apk/res/android"
        xmlns:tools="http://schemas.android.com/tools"
        xmlns:app="http://schemas.android.com/apk/res-auto"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        tools:context=".map.MapsActivity">

    <fragment android:id="@+id/map"
              xmlns:tools="http://schemas.android.com/tools"
              android:layout_width="match_parent"
              android:layout_height="match_parent"
              tools:context=".map.MapsActivity"
              app:layout_constraintTop_toTopOf="parent"
              app:layout_constraintStart_toStartOf="parent"
              app:layout_constraintBottom_toBottomOf="parent"
              app:layout_constraintEnd_toEndOf="parent"
              android:name="com.google.android.gms.maps.SupportMapFragment"/>

</androidx.constraintlayout.widget.ConstraintLayout>

The user will be prompted to accept the Location permission:

The location will be updated on app launch, and every two minutes:

Extra note regarding AndroidX

If you are using AndroidX, you might need to add these lines to your gradle.properties file (see here for more info):

android.useAndroidX=true
android.enableJetifier=true