What are some good PSP games that I should consider getting?
 
I'm looking especially towards JRPGs and fighting games (2D/3D), but I have no real idea.
 
My library at this point is:
Ratchet and Clank: Size Matters (bundle)
Echochrome (bundle)
Lumines II
Tony Hawk's Underground 2 Remix (borrowed and friend didn't want it back)
Gran Turismo
Persona 3 Portable
DJ MAX Portable 3

street fighter alpha 3 max

Kenka Bancho Badass Rumble
ZHP
Disgaea
Lumines
Lunar Silver Star Harmony
Half Minute Hero

Disgaea 2  
 
Unlosing ranger 
 
Metal Gear Acid 2 (if you can find it) 
 
Blazblue portable 

" street fighter alpha 3 max "

There are 2 really good God of Wars on the PSP and also 2 really good Tekkens (stay away from the really badly done Soul Calibur).  For JRPGS I'm not a fan so I can't recommend anything.

kingdom hearts birth by sleep 
 
only reason i play my psp

" kingdom hearts birth by sleep  only reason i play my psp "

Do you like PAIN? 
 
If so check out Dracula X Chronicles, guaranteed to make you want to pull your hair out with its old school difficulty. Or if you're like me you could just unlock Symphony of The Night, play that and not finish the main game until around two years later. I still can't believe I went back and finished that game.

I say grab Valkyria Chronicles 2

ZHP

It's not a JRPG or fighting game but Crush is criminally underrated and has some of the best level design you'll play.

peace walker,  valkyria chronicles 2, ghost of sparta

Hm, so it sounds like KH:BBS is a must get. Anything else?
 
How well will I be able to understand its story given that I've only played Kingdom Hearts I to completion and that was during its year of launch aeons ago?

a second thumb stick. 
 
nm i just saw the part about fighting or rpg. 
 
valkyria chronicles 2 then. or that monster hunter game

Wipeout Pure or it's sequel: those games are pretty good, even if you don't really like racing games 
I think some of the Valkyria Chronicle games are on the PSP - I haven't played them, so I don't know how good they are, but they are JRPGs and they received good reviews, so you probably want to think about. 
 
I would avoid Metal Gear Solid Portable Ops - the controls on the PSP, frankly, are garbage. I heard Peace Walker had substantially better controls, so you might want to pick that up.  
 
If you like puzzle games, Echochrome is great - it was on UMD here in Australia and anywhere outside the US. Fun Stuff. 
Persona 3 Portable is good. I'm guessing from your avatar that you like the Persona games, so it's worth a shot. 
 
I heard the God of War game was also good. Roco Loco was also excellent.  
 
Stay away from Tokobot - well, I found that game to be bad at least. 
 
Final Fantasy Tactics is a good choice if you haven't played the original FF tactics.  
Metal Gear Ac!d - ehh.... not bad. But only if you can get it super cheap. 
 
Funnily enough, out of all my PSP games, I've enjoyed Lumines the most. And that was the first one I got.

I highly HIGHLY recommend Burnout Dominator. It's a perfect combination of Burnout 2 & 3. 
 
Tekken 6 is damn good as well, especially since all the fighters are unlocked from the start.
 
Tales of Eternia is  a really good rpg, the 2D visuals look damn good on the psp.
 
Me And My Katamari - gotta have some katamari action there, it does work suprisngly well considering the PSP control layout.
 
Also I really enjoyed Undead Knights. The graphics can be a bit janky but the whole mechanic where you turn your enemies into zombies and then fight on your site is really satisfying.

The new God of War and the older one if you can find that one also.

" ...If you like puzzle games, Echochrome is great - it was on UMD here in Australia and anywhere outside the US. Fun Stuff. Persona 3 Portable is good... Funnily enough, out of all my PSP games, I've enjoyed Lumines the most. And that was the first one I got. "