Racing Games;
Need for Speed Most Wanted is the latest installment for the NFS series. A nice game, targeted for those "The Fast and The Furious" fans.
If you're more into the serious racing games you should definetly go for Gran Turismo 4, which lasts really long since there's literally hundreds of cars and loads of racing modes available.
Fighting Games;
The Tekken series are really good fighting games, more hand to hand combat instead of swords and other weaponry.
Virtua Figter games are also very nice, although they are more difficult to master than Tekken games since the buttoncombinations for various moves in Virtua Fighter are harder to do.
Sould Calibur is pretty cool if you're looking for a fighting game with weaponry. And if you're into more violence than you could always have a look at the Mortal Kombat series.
RPG's
Final Fantasy games are good, at the moment you have only 2 available for PS2 (Europe) FF X and FF X-2 (stay away from the last one unless you like girlie stuff and dressing up ;D) The up and coming FF XII for the PS2 seems to be a really great game from what I've seen of it so far. You could always try out the older installments of ther FF series since the PS2's backwards compatible with those, although the graphics are outdated some parts have storylines that exceed the PS2 versions.
Kingdom Hearts 1 and 2 are really nice RPG's with great storyline, but might look too childish since there's the whole Disney cast running around in there.
Dragon Quest and Tales of Symphonia are also really nice. They're both Cel shaded games like Zelda the Wind Waker for the Gamecube, which you'll either hate or love.
1st person shooters;
If you've been spoiled by FPS games on the PC you'll have a hard time getting used to the controls and the lesser quality of graphics of 1st person shooters on a console, but rhere are some;
Killzone is a cool futuristic themed shooter with good graphisc and storyline, it (as mentioned before) takes a while to get used to, since it'll look a lot less fast paced than other shooters.
Timesplitters 1,2 and 3 (Future Perfect) are a lot more fluid than Killzone, and offer a nice storyline but their strong part is the amount of extra gameplay modes they offer. Not so serious as Killzone since it involves monkeys and walking trees and other weird charaters to play as.
Maybe other HWF users have more suggestions, I hope this helps you a little. Most of the games I suggested aren't that expensive anymore nowadays.
Greetings,
Marcus