I think the PSP supports MP4 and AVI. It doesn't matter that much which version you buy, even though I have one myself I wouldn't recommend the PSP Go as it's a download only machine and the prices on the Playstation Store can sometimes be a bit pricey, so that leaves you with one of the candybar style PSP's. There's the 1000 which was the first model and doesn't have a microphone, it also has the worst screen, the 2000 is when they made it a bit slimmer and added a built in microphone as well as improving the screen and then I think the 3000 offered more improvements to the screen and more lightening/slimming.
The 3000 is the one you'll typically see in the shops and I think it can be had for around £100 new. It depends how much money you want to spend on it, you can pick up second hand PSP's fairly cheaply and there's often ones up for sale in the MM. To play PS1 games you download them from the Playstation Store, you can do that directly from your PSP, or you can use a program on your PC to do it. The best games I'd say would be the games in the Metal Gear Solid, Tekken and GTA series' and Gran Turismo, but there's loads of RPG's and stuff like that, just look on your website of choice for reviews.
Some of the typically high scorers are:
Metal Gear Solid Peace Walker
Patapon
Burnout Legends
Lumines
Final Fantasy Crisis Core
Wipeout Pure
SOCOM Fireteam Bravo
Tactics Ogre
Daxter
