I'd advise against using processing to make a game, it's not really built for it. It lacks a way to test for the status of a key, you just get events. Things like joypad support and sound are handled through external libraries. The accelerated graphics support is still pretty dodgy.
Slick 2D is a much better starting point. It's an object layer over OpenGL using the LWJGL library (so it's accelerated by default). It has game specific stuff like classes to handle tile based maps and A* path finding.
If you want to build your own engine,
LWJGL is (like Tao is for .Net) a simple access layer for OpenGL, OpenAL and some input/controller stuff for Java. Tao has some other neat libraries aside from those, but that's the meat of what you'll need for cross platform game development using accelerated graphics anyway.
It seems LWJGL is a bit better maintained than Tao. It already has OpenGL 3.0 support while there hasn't been an official Tao release since May last year.
I've been using LWJGL for quite a while now and I'm very content with it. I like Eclipse a lot better than Visual Studio and Javadocs a lot better than MS's horrid documentation thing. Also, you can make
webstarts with Java, so people on any platform can play your game with one click.
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