Use 12.04 unless you really like trying out the newest version.
I'm surprised that no one has asked you yet. How well do you know Linux? What experience do you bring? Have you tried Unity before?
If you are a very new user, coming from Windows, then the answers might be different. I would say try Cinnamon Remix, which is newbie friendly, avoids Unity and is based on Ubuntu 12.04 for it's long term support.
Momist. Follower of the minor Greek god Momer.
If you want a system that just works and is supported a long time, go with 12.04. If on the other hand, you prefer being on the more on the edge, then go with 13.04.
13.04 is a reasonably conservative release and if you are just 'testing the waters' I would have a look at it. Unity seems a little more polished on this release as well...
You think that's air you're breathing now?
I also heard that Chrome doesn't work on 13.04 if you just download and install. You have to download another file/program to make it work properly?