I would advice to start by trying & playing with standard network object (Sockets) than once you understand how it work you may want to switch to a library (as it is IMO always important to understand how things works).
Also start with TCP it is a lot easier, you should be able to build you first client/server pretty quickly with a good tutorial
I second all of this advice. Always start at the bottom and work your way up. Once you understand how basic sockets work, then you can decide whether you want to (a) keep using them, (b) learn a new API, (c) write your own library, (d) switch to someone else's library. TCP is definitely the way to go for almost all applications. Certainly read about UDP in order to learn the differences, but focus initial coding on TCP sockets.
java.net.Socket and java.net.ServerSocket will get you along way toward your goals.