Pseto,
The PSP comes with its own browser that works much like Internet Explorer or whatever you use. It has bookmarks, forward page, back page, home, etc. and an innovative method of keying in the address. For example there is a button that types in "http://www.", ".com", etc.

As for connecting you will need to have a wireless access point. The device will automatically search for a hotspot near you. You would be able to surf the net in an airport for sure as I believe nearly all of them have wireless access points. You wouldn't be able to on a plane though I don't think . But you can still listen to music, watch videos or movies, or play video games.