Okay. PSP games...where to start?

For action RPGs, I'd say the mainstays are Final Fantasy 7: Crisis Core (one of the most impressive handheld games I've ever seen in terms of production values), Monster Hunter Freedom 1&2, and Bounty Hounds. Bounty Hounds is actually very reminiscent of Phantasy Star Online, though it does suffer from a *somewhat* problematic camera.

It may also interest you to know that a revamped version of Phantasy Star Universe is coming to the PSP as well (and I believe the single-player adventure is brand new).

Tales of the World: Radiant Mythology is also a great action/active RPG which features an ensemble cast from all the Tales of… series. I've had a great deal of fun playing it.

An official sequel/prequel (I forget which) to Kingdom Hearts is also coming: Kingdom Hearts: Birth By Sleep. It looks pretty amazing.

For SRPGs, you really can't go wrong with the following:

Final Fantasy Tactics: The War of the Lions - I can't stress enough how vast of an improvement this is over the original. The new translation does wonders for the story and the cut scenes are simply gorgeous. New characters as well as multi-player battles really add a lot to this classic.

Jeanne D'Arc - Level 5 at the top of their game. Might seem a touch on the easy side if you're a real strategy nut though.

Disgaea: Afternoon of Darkness - This update of the PS2 original improves on it in virtually every way (even in the technical department). There is a lot here to sink your teeth into…the Item World is an amazing addition and "Etna Mode" allows you to play an alternate story which imagines a "What if…" scenario where Etna kills Laharl.

Wild Arms XF - A Wild Arms SRPG which very much follows in the footsteps of Final Fantasy Tactics. Great fun and a good challenge coupled with excellent character designs and beautiful artwork/music.

Silent Hill Origins should absolutely not be missed. This game is impressive. Period. The graphics are gorgeous and atmospheric, the story ties in nicely with the plot of parts 1 and 3, the music is great, the lighting effects are better than a lot of what I've seen on the PlayStation 2, tons of extras as well as its own "achievement system" w/ rewards. This is one game where I simply don't understand the criticisms levied against it as its easily one of the most technically impressive games on the PSP and has the most refined combat/item systems of any Silent Hill to date. I also found the atmosphere to be just as effectively creepy as any of the console versions.

Some other games which I feel really set the bar on the system or are simply good fun:

Tekken: Dark Resurrection
Power Stone Collection
Me and My Katamari
Crush
Exit
Loco Roco
Patapon
Ridge Racer
Wipeout Pulse
Crazy Taxi: Fare Wars
Ace Combat X: Skies of Deception
Megaman: Powered Up
Megaman: Maverick Hunter X
Castlevania: The Dracula X Chronicles
Daxter
Ratchet and Clank: Size Matters
The Warriors
God of War: Chains of Olympus
Metal Gear Solid: Portable Ops
Gurumin: A Monstrous Adventure

There are also a myriad of good, more traditional RPGs and some of the best retro compilations to be found anywhere. I hope this list wasn't too exhaustive, and just let me know if you want a bit more detail on any of the games I've mentioned.