Get a solid understanding of Java, then the Servlet API, then Spring. Once you have those three, you have a pretty solid foundation for the majority of Java development I've seen and dealt with. A bit of Hibernate / JPA in there wouldn't hurt either.

oh wait...

On a professional level I'd advised EE everytime - it's the most common requirement listed (at least here in the UK) next to Java for applicants. On a personal level, I'd say don't specialise, enjoy instead. JavaFX looks cool, scripting is cool. Find something you like and do that.

The main problem with Java programmers (and many people in many industries to be honest) is that they are overly specialised. They're essentially one-trick horses that can't do anything but that one thing. Professionally this will eventually lead to a dead end when (not if, when) your chosen technology is no longer "popular". You end up doing eternal maintenance on ever older code bases until those are finally replaced with other products created using tools you never saw, making you a dinosaur and ending your career. Generalise, don't specialise. Of if you specialise, do so without loosing sight of the larger environment.