I try to be as diverse as I can and try new things as much as possible. I've tried to try out every franchise in the Smash Bros series, and I have done so thus far. I can't think of any that I dislike, except for Mario. I like some games, but the spinoffs get obnoxious and Galaxy wasn't even half as good as the past two big games (64 and Sunshine). The only franchise that has a rep in Smash Bros that I have not tried is Metal Gear. I'd like to try it out, but on the other hand, I don't have a PS3 or a PSP, nor do I think I will get one. And if I get really into the series...well, I'm going to want to spend $600 (or however much it costs now) on a PS3 and eventually buy the PSP, which has about 3 games that I'm interested in. I already have a Video iPod so I don't have much other use for it.
In any case, my favorite genres are probably adventure games and turn-based RPGs (except not Final Fantasy (I know, blasphemy), Fire Emblem and Pokemon mostly; Earthbound, or at least what I've tried of it, is pretty good too). But I'm really open to just about anything.
The only genres I'm not a huge fan of are FPSs and Sports. I don't like sports in real life all that much, so I'm not a huge fan of them in video games. FPSs are generally pretty boring to me because for the most part, you've played one of them, you've played them all. These days, people rave over games like Halo which bring nothing new to the table. It's sort of an overrated genre altogether, in my opinion. The only games I'm a big fan of that are sort of FPS are the Metroid Prime games (which Nintendo has defined as First Person Adventure, which actually makes sense, as stupid as it sounds).