The previous users have pretty much summarized well the games you should get.
· Lost Odyssey should be your top priority. After that, Tales of Vesperia, Nier (if you didn't play it on PS3, since it plays better on 360) and The Last Remnant. There's also Blue Dragon, Magna Carta 2 and Infinite Undiscovery, which aren't particulary engaging, but no harm in you giving them a try.
· There's a bunch of shumps that are extremely fun. My personal favourite is Akai Katana, but don't miss out Doddonpachi, Deathsmiles, Ikaruga and Radiant Silvergun. (Most of these can be found in the marketplace, but there's several physical versions available).
· Some strange japanese games, like Ketsui, Senko no Ronde, Culdcept Saga, Vampire Rain or Operation Darkness. These might be particulary hard to track (some were USA only), and I can't attest for their quality. It's worth giving them a shot, though.
· Best Halo available is Halo: CE Remastered. That being said, you can get yourself a copy of Halo 2 (for the OG Xbox) and it works like wonders on the 360, plus four-player splitscreen available to you right there. Might as well try all Halo games on 360, although I personally hate Reach.
· The Fable games; despite the first one being on PC, the third one is broken and can't be played anymore (GFWL), and the second always remained 360 exclusive. Second one is the best, so there's also that. It's not heavy on RPG mechanics or anything, but it's a fantastic and a charmer of a game.
· Gears of War is hella fun, probably the best flagship MS franchise on 360 (yeah, even better than Halo in my eyes). Even if the macho steroid hero bravado saving the world from ugly monsters isn't really your thing, just go ahead and give it a go. It's great in co-op (splitscreen), and has plenty of replay value and interesting extra modes.
· Project Sylpheed, to satisfy that need of a good Star Fox game that never came during the seventh gen. It's a short game, but very sweet, especially if you're into space combat.
· There's a bunch of extra games you could see yourself getting. Beautiful Katamari is fantastic, so go for that. Ace Combat 6, if that's your jam. Crackdown games are okay, I personally didn't like them, but they are fun for some people. The Diswasher (digital only) can prove to be engaging. But above all, don't miss out on Geometry Wars 2 (also digital-only); it's a fantastic game and extremely engaging, plus can allow up to four people playing it on the same screen. You don't even need to play the original one since this one is basically the original one but revamped, with more modes. Do yourself a favor and get this ASAP.
· Some Rare games, like Kameo: Element of Powers, or Perfect Dark Zero. They aren't particulary great, but can be enjoyed.
· Rumble Roses XX, Onechambara: Bikini Samurai Squad and Dead or Alive Xtreme 2, if you're feeling horny. Hey, there's tastes for everything!
· Some multiplatform games that were under the radar and performed better on 360, like Majin and the Forsaken Kingdom, Condemned (given that only 360 holds both titles of this franchise), and the likes.
EDIT: They were ported to everywhere else at this point, but goes without saying, the Dead Rising games. There's a Xbox 360-only collection that actually holds two titles (Case Zero and Case West) that are 360 exclusive, and you get over five fantastic games, probably among my favourites on 360. Get them too!
Pretty sure I've mentioned over 15 games, so you have plenty to work with already. Hope this is sort of what you were looking for.