There are some great game you can get on 360.
The Assassin's Creed franchise is really good so do try them
Bioshock is a fantastic game too. Great story and atmosphere
The Orange Box is a great package as well
The Batman games are really good too: Arkhan Asylum and Arkham City
If you like huge open world RPGs then Oblivion or Skyrim should be good for you
Any LEGO games are really, really fun. It doesn't matter if it's Star Wars, Harry Potter, Batman or Indiana Jones, thy're all really fun
Mass Effect is a great sci-fi RPG series
Fallout 3 is another great open-world RPG
There are a lot of great games on the Xbox Live Arcade too
Braid is a wonderful puzzle game
Resident Evil 4 is a thrilling action game
Castle Crashers is a really fun hack-n-slash game, especially if you have some people to co-op with but it's great alone too
Portal is an excellent puzzle game (and is avaliable as part of The Orange Box but also avaliable by itself on XBLA)
N64 classics Banjo-Kazooie and Banjo-Tooie are also on the XBLA.Excellent 90s era 3D platformers
Your comment about Call of Duty makes me a little wary of suggesting Halo because I get the impression you don't like competetive shooters but Halo is very, very different to CoD. It has a great single player and great multiplayer, if that's something you enjoy. I would highly recommend Halo but if online shooters aren't your thing then you probably won't get into it.