Well, yes you can browse the internet freely on your PSP through the use of a wireless access point since it has wi-fi.
What you heard about the "3004" model is bullshit - you can do that with any PSP. The only way you can play games off your memory stick is if you download them from the Playstation Network Store (don't, they're a closed market and charge whatever they want) or if you get your PSP hacked to run custom firmware (harder to do on a 3004 model).
The PSP is probably the most easily hacked console to exist in recent years, and Sony are incredibly pissed about it. If you're thinking of getting it hacked, I'll have to tell you that it's easier to do so on the earlier console models, e.g. 1000 - 2000 series. But discussion about piracy and such isn't allowed here so that's as far as I'll go.
As for the PSP itself, I've had mine for 5 years now and it has served me well. It's good for listening to music (but too bulky to use purely as a mp3 player) and watching movies on the go, and there are some great games on it.
You can also play old PS1 games on it if you download them from the PSN for a price/get a hacked console and copy the games from your old PS1 discs.
The controls can sometimes be awkward for some games, especially FPS games which use the d-pad to control the camera, leading to having to use the "crab claw" technique, involving your index finger controlling the d-pad and your thumb on the analogue stick.
tl;dr:
Yes the PSP can browse the internet freely provided it can access a wireless access point.
PSP is easily hacked for those inclined.
Good for a variety of uses (music, movies, games, photos)
If you're planning on hacking it, get an earlier model (1000 series or 2000 series)
FPS games are awkward on PSP