I had the same issue. I get a date as a String, for example: '2016-08-25T00:00:00', but I need to have the Date object with correct time. To convert String into object, I use getTimezoneOffset:

var date = new Date('2016-08-25T00:00:00')
var userTimezoneOffset = date.getTimezoneOffset() * 60000;
new Date(date.getTime() + userTimezoneOffset);

getTimezoneOffset() will return ether negative or positive value. This must be subtracted to work in every location in world.

Datetime Documentation is actually confusing.

Note:

valueOf: Returns the datetime in GMT timezone
valueOfGmt: Returns the datetime relative to GMT timezone

So, change the code to below:

Datetime today = Datetime.now(); 
String myDate = today.format('yyyy-MM-dd HH:mm:ss');
Datetime myDateTime = Datetime.valueOfGmt(myDate);

System.debug('myDate:: '+myDate );  
System.debug('myDateTime:: '+myDateTime); 

Assuming you are referring to the part of Florida following EST, you can set the timezone for SimpleDateFormat and set your TimeZone to EST.

SimpleDateFormat format = new SimpleDateFormat("yyyy-MM-dd HH:mm:ss");
format.setTimeZone(TimeZone.getTimeZone("America/New_York"));
Date date = format.parse("2014-01-30 07:48:25");

Your parsed date now can be utilized by your default TimeZone of the system (or set it to your liking as we did in the first place).

TimeZone.setDefault(TimeZone.getTimeZone("UTC"));
System.out.println(date);

The output I get for your date offset to UTC:

Thu Jan 30 12:48:25 UTC 2014