a huge part of the Xbox 360 is online. most of their games are/will be geared towards multiplayer.
i dont think i can recomend many games, as i mostly play multiplayer ones.
and i think our taste in games in general is difrent.
Ghost Recon Adv warfighter is a good game, but its replay value is not so hot.
Oblivion is a great game, but its also on PC. so i donno where you would stand on this one.
Saints row is coming out soon, it will be the 'GTA' for Xbox. it seams pretty good single player. but it is practicly the same game as GTA, only evrething from your clothing to your car is customizable. it has missions sort of like GTA2 had, with the gangs at war. and picking your side.
it boasts online Multiplayer, gang Vs gang. ( Planetside style? i doubt it, but it would be sweet )
my definite advice is rent befor you buy. i have saved myself a TON of cash this way.
also, if you want to avoid the multiplayer... AVOID Chromehounds. single player in that game is stictly training for the online portion of it.
even if you dont want to use the Multiplayer hook up your xbox to the internet anyway, their is always TONS of stuff to download from upcomming games. Vidio's Demos, pictures. ect.
edit - also avoid Ninighty nine nights... i have heard nothing but bad coments on it from evrebody whos bought it.