Hello to you Yuna! I'm new myself (joined a couple of days ago)
To answer your question, my favourite genres are RPG, Adventure, Tactical Espionage, Action and FPS (with puzzle solving elements in each genre).
My favourite all time game being Deus Ex, because it's a mix of RPG, Adventure, FPS, Stealthy Tactics and more. Plus, its really fun, has alot of freedom and you can play through the games and experience new things (your actions have consequences and alter the outcomes of the game). It's also a long game. (I can't wait for the sequel in March 2003 (I think))
Second would have to be Abe's Exoddus because it's long, challenging and fun. Abe's Oddysee follows that because it's fun and challenging like Abe's Exoddus, but its not as long. Both games had beutiful 3D rendered backgrounds and characters, and the level design was just superb (with advertisement of products scattered around the industrial levels in LED signs and pictures and other features made it all seem like a believable place).
I'm not sure about what comes next. But in no particular order, I also like the following games...
Halo - excellent game, better than half life in some aspects, not as good in others. Superb AI and enemies are fun to kill
Half Life - a classic FPS with superb level design and again, fantastic AI.
Fallout 1 & 2 - classic RPGs with alot of freedom set in a post apocalyptic world. Turn based combat and very fun.
Munch's Oddysee- Not as good as the original oddworld games, but still very fun.
Doom 1 & 2 - addictive shoot 'em up, very fun. nuff said
Diablo 1 & 2 - a bit repetitive after you've completed it, but a fun hack 'n' slah action adventure
Spiderman: The Movie -web slinging is fun
and theres much more