Skyrim: It has hundreds of hours of gameply, it is a Bethesda game so your game can be different every time, and it has such an interesting story.
Call of Duty Modern Warfare 3: It is a fun game to play multiplayer on, it has had a plethera of improvements since the last modern warfare, it is a great game to releave stress on, and it's a game you can easily play with friends.
Red Dead Redemption: Even though this game is a little old it still holds up VERY well, it makes use of Rockstar's amazing creativity, it has an engaging store, and the multiplayer is so much fun to play with friends.
Saints Row the Third: It is such an wacky game that you can't stop smiling while playing it, it's co-op is over the top, and did I mention how whacky it was?
Fallout New Vegas: Another Bethesda game and it does not disappoint, it's story is so interactive, it has so many different gameplay aspects, and if your going to get a dlc for it, get old world blues, it has to be one of the funnies dlc's I have ever played.
Those are just of few of my favorite games, hope it helps.
Some good games for Xbox 360 include Gears of War 1 and 2, which are nice FPS games. Others include Mass Effect and Final Fantasy XIII, which is created by Square Enix that has great experience in creating RPGs.