As an add-on to the accepted answer: to install a certain version of Java, e.g. version 11, run:

brew install openjdk@11

And symlink it:

sudo ln -sfn /opt/homebrew/opt/openjdk@11/libexec/openjdk.jdk \
     /Library/Java/JavaVirtualMachines/openjdk-11.jdk

I have been doing some research on how to install Java jre and jdk on correct PATH.

JRE has been discontinued since Java 11. Since you are talking about Java 14, there is no JRE for it. You just need to install (and configure, if required) JDK.

Use the following command and then check java -version:

export JAVA_HOME="/Library/Java/JavaVirtualMachines/adoptopenjdk-14.jdk/Contents/Home"
export PATH="$PATH:$JAVA_HOME/bin"

Note that it will work only for your current session i.e. java -version won't work in another terminal window.

To set JAVA_HOME permanently, do as follows:

$ cd ~
$ vi .bash_profile  

Write the following line into .bash_profile file, save and quit:

export JAVA_HOME="/Library/Java/JavaVirtualMachines/adoptopenjdk-14.jdk/Contents/Home"
export PATH="$PATH:$JAVA_HOME/bin"

Then refresh (read and execute .bash_profile)

$ source .bash_profile

And finally test

$ echo $JAVA_HOME
$ java -version

Note: Also, This thread may be useful to you.