Mix-and-match, really.
Right now I'm playing Oblivion because it's a very decent rpg that lets me bash in skulls with a giant warhammer, set things on fire and shove a very sharp piece of steel up someone's nose. Yes, I've had a pretty bad day, why do you ask?
Also, I just finished assassin's creed, and I loved it, open-world action where the combat didn't suck and the sneaky-runny-jumpy-climby was awesome. I'm impressed, especially since I love open-world games and especially those where you can dick around on rooftops.
Mass effect 1 and 2 I've played through time and time again since they're damn good rpg's and otherwise good games, more than competent shooters.
Halo 3, because I love the multiplayer and it's really not a bad game in single-player either, plus I like that they drop all pretensions of realism.
Dead or Alive 4 I bought to see if I still kicked ass. That, and it's not a bad fighter.
Thus is my 360 catalogue summed up, minus Bioshock, which I just could not get into.
Why I play these games? A variety of reasons, relaxation and a feeling of accomplishment among them.