Any other suggestions (hardware, etc.)? Other game suggestions (while I am primarily a shooter/MMO fan, I play a little of everything if its good)?
Borderlands, Fallout, Crackdown, Mass Effect 2, Geometry Wars 2 (XBLA).
Borderlands is a loot-based shooter and crazy awesome. It is fun single-player, but can also be played 4-player co-op. Lots of fun.
Fallout is real hit or miss with people. I love it, but I have friends who don't. It is an RPG Shooter. Both Fallout 3 and Fallout New Vegas are quite awesome, and very similar. If you can find Fallout 3: Game of the Year edition cheap, grab it. It has the game plus all 5 of the DLC included. Lots of hours of play in that one. Single player only. New Vegas is still quite new, so it probably isn't cheap yet, but it is a great game that I wouldn't hesitate to recommend.
Crackdown is a 3rd person action game. Think GTA, but reversed. You are a cop with super abilities, and as you progress, these abilities get more powerful. The story is crap, but it is super fun action, and can also be played co-op through the campaign. Crackdown 1 and 2 are very similar. I personally feel that CD2 fixed a lot of the issues from CD1, with next to no downsides. So, if you can find CD2 cheap, get it. If not, CD1 is still awesome and can probably be found for next to nothing now a days.
Mass Effect is a 3rd person action/RPG. The first one was highly acclaimed, but I never played it. Mass Effect 2, though, is totally amazing. The story, the characters, the action, the sci-fi environments, etc. All of it is quite an awesome experience. Plus, you can play the game in many different ways, so multiple playthroughs can have
VERY different results.
Geometry Wars 2 is one of my favorite games this generation. It is on Xbox Live Arcade. It is a twin-stick shooter. Think Asteroids, on speed, turned up to 11. Lots of different modes to play, all of which are quite different and fun. Plus, it uses XBL's leaderboards to great effect, by taunting you with your friends scores in all the modes if they play. Awesome fun.
As for internet speeds, the speed isn't really the issue, it is your ping time that matters most. You can play just fine with as little as 1mbps, as long as your ping times are low enough. The higher your ping times, the more lag you will notice in online games.