There’s lots of things can break when changing versions of ruby and rails. If you’ve got good test coverage this will obviously be easier. I’ve found ruby version changes to be significantly less painful than the jump from rails 2.x to 3.x, so I’d suggest starting there.
I would upgrade your rails 2.3.16 app to Ruby 1.9.3 at the outset. The jump from there to 2.0 will probably be completely without issue as there are very few backward incompatibilities. I’d say just go straight to 2.0, but only the most recent version of rails 3.2 supports that. So you’ll have to wait on that.
Upgrading to rails 3.x is daunting. I’ve read several places that it’s likely best to go from 2.x to 3.0.x and then to 3.2.x. 3.1 introduces the asset pipeline and you probably don’t want to deal with that on top of everything else. I recall seeing high praise for this guide, but I have not tried it myself. There are likely other guides available as well.