Back when Emacs moved from Bazzar to git, Lars Ingebrigtsen wrote a short introduction to Emacs development. I meant to write about it but lost track of the post. Happily, I've found it again and am posting about it to encourage any hackers with a few spare cycles to take part.
It turns out that the Emacs bug tracker has a wish list so you can jump right in enhancing some existing feature. Ingebrigtsen's post tells you how to get the Emacs repository, compile Emacs, and access the wish list. You can, of course, also tackle some of the bugs if you'd rather.
The worst part is executing the FSF copyright transfer. It's much fussier than it should be but not really hard. You just email in a form, electronically sign the contract that you get in return, and email it back. You can also do everything by snail mail if you'd rather. Do it right now, though, because there's lots of lag time and when you have a patch ready to go it's very frustrating to be sitting around waiting for the FSF to do their thing.