I generally prefer my 2000 model PSP, though I've one instance of a compatibility issue. Disgaea 2: Dark Hero Days will run on it but doesn't work properly. It works fine on my 1000 model though, so there's nothing wrong with my copy of the game. From what I've read, there's a couple different versions of the 2000 which have different motherboards depending on when they were made. Perhaps other 2000 model PSPs with the different motherboard wouldn't share this compatibility issue. Aside from that, I had to tape down the switch that the system uses to tell if there's a UMD inserted because it wasn't being pushed in all the way when I actually had one in. That solution has worked just fine and I've had no issues since. Other than that I haven't had issues with it.
The 1000 model is sturdier, but bigger and heavier. That could be a good thing, depending on your personal preference. I haven't used it as much as my 2000 model, but I haven't had any issues with it either.
I haven't used the 3000 model or the PSP Go, so I can't really comment on either. The PSP Go only plays digital copies of games you have on PSN, so I can't really recommend that to anyone as their first PSP.
In terms of cost, there's not too much difference between the 1000 and 2000. You should be able to get whichever one you want in one of the standard releases for under $60. They're both nice, either will work just fine.
