It's tough to recommend games for someone who has yet to venture far from Nintendo consoles...
You like F-Zero a lot, which means you can tolerate racing games, which means I recommend something from the Burnout series. I think my top three favorite Burnouts are Burnout Paradise, then Burnout 3, then Burnout Revenge. You might go for Burnout 3 since it's an Xbox Original game, so it'll be much cheaper.
Most of the games people have listed here are really good, but I'm not sure what kind of a gamer you are, so I don't know if you'd like them. Keeping in mind your terrible status for online gaming, I'm going to recommend good single-player games. Assuming you'd be willing to play any type of game, my top three recommendations out of every Xbox 360 game ever are BioShock, skate., and Guitar Hero 3.
As for Live Arcade, yeah, you can play the free demos, so just try them out before buying.