IMHO, if you are going to play a Devil May Cry game, you might as well skip right to Devil May Cry 3, it's a prequel so you aren't missing out on any backstory and the combat system is just so much more fun than the original.
The PS2 holy trinity of action platformers is Jak and Daxter, Ratchet and Clank, and Sly Cooper and the Thievious Racoonus. I recommend you purchase the first game of all 3 of those series and give them a playthrough. Based on your experiences from the first games you will pretty much know which series you will wish to continue.
Another vote for the weird and wacky Katamary Damacy.
R-Type Final is a great old skool style 2D ship shooter
The Socom series is a great tactical shooter, but it is most fun when going online.
If you can track it down, Zone of the Enders: the 2nd Runner is an outstanding and graceful mech combat game.
What other systems do you have besides PS2? If not an Xbox, pick up a Grand Theft Auto.
