LibGDX is my weapon of choice, note though that many of the tutorials out there are set up pre-0.9.6 and quite a lot got changed in that update, rendering most of them useless. https://www.youtube.com/user/doctoriley has a decent series of tutorials which should give you a starting point (not exactly best practice but you get to understand how it all hangs together) and the code can be found at https://bitbucket.org/GQGK/creating-a-game-in-java-youtube-series/overview if you want it.
LibGDX is excellent, a layer on top of lwjgl which provides support for Android, desktop (win/lin/mac) and html5 (though frankly I'd say just use an applet instead as it's quite limited) and I believe work-in-progress iOS support too.
The documentation is relatively solid (though the wiki is incomplete and outdated) and it has been used in commercial projects.