I have had my xBox 360 for a month now and the best game I have for it is Oblivion. The other games I have are okay but just about every game available is either a FPS or a driving game. The best thing about it is the XBox live arcade which is the main reason that I got it, as a few of my friends have one and we are getting very competative about Pac man, Galaga, Marble Blast and Geometry Wars.
Some good XBox games (which are cheapo these days) that work on it are Halo, Serious Sam and Half Life 2 (again, all FPS's).