Survival Horror:
-Silent Hill 2 (Director's Cut or no)
-Resident Evil 4
Shooter:
-TimeSplitters - watch out though, really old school.
-TimeSplitters 2
-TimeSplitters: Future Perfect
-Red Faction
-Red Faction 2
-Killer7
-Transformers
Platformer:
-Jak & Daxter
-Jak II: Renegade
-Jak 3
-Ratchet & Clank
-Ratchet & Clank 2
-Ratchet & Clank 3
-Ratchet & Clank Gladiator (or Deadlocked in the US) - my opinion, not as good as its preceders.
-Sly Raccoon
-Sly Raccoon 2
-Slay Raccoon 3
-Rayman Revolution - watch out though, really old school.
-Rayman 3: Hoodlum Havoc
-Maximo
-Maximo And The Army Of Zin
-Psychonauts
Action/adventure:
-Devil May Cry
-Devil May Cry 3 (Special Edition or no)
-God Of War
-God Of War II
-GTA: San Andreas
-ICO
-Shadow Of The Colossus
-Viewtiful Joe
-Viewtiful Joe 2
-Zone of the Enders
-Zone of the Enders: The 2nd Runner
-Okami
jRPG:
-Final Fantasy X
-Final Fantasy XII - Maybe, if you want something that looks a bit more like an offline MMO.
-Shin Megami Tensei: Lucifer's Call - Offers a challenge and is a MUST for every self-respecting jRPG player.
-Shin Megami Tensei: Digital Devil Saga
-Shin Megami Tensei: Digital Devil Saga 2
-Shin Megami Tensei: Persona 3
-Persona 4
-Shadow Hearts - Watch out though, really old school.
-Shadow Hearts: Covenant - Want a good jRPG? Here you go!
-Dragon Quest: The Journey Of The Cursed King - Contains many old school elements.
Puzzle:
-Bombastic
-Wetrix 2.0: Aqua Aqua
Music/misc.:
-Rez
-freQuency - not for everyone.
-Amplitude - sequel to freQuency, so not for everyone either.
Beat 'em up:
-Soul Calibur II - SC III too, I suppose, if you really dig it, but it's a big rehash.
-Tekken 5 - Tekken Tag Tournament and Tekken 4 too, I suppose, but they're really the same, just worse.
Racing/car combat:
-Burnout: Revenge
-Twisted Metal Black
-Mashed (Fully Loaded or no) - Play with friends or don't play at all.
What I would recommend from my collection.
Make sure to check out 2nd hand games (in stores or online), bargain bins and platinum titles. With a bit of luck and financial insight, you can get 6 PS2 games for the price of one PS3 title.
Also, I don't play sport games except for the ol' Tony Hawk games, so no advice on that.