Networking is a layered architecture, debug in layers, starting at the bottom.
Can you see network card? Good!
Can you connect? Good!
Can you ping your peer? Good!
Can you ping elsewhen? (Using an ip number) Good!
Can you ping elsewhen using an ipname? Good DNS is working.
Can you use a simpler facility to access a web site. eg. Use the "net cat" facility.
You should see a stream of html coming at you. If you do, great! If you don't. Aha! Maybe you have a firewall playing silly buggers. Switch the @%$#! thing off.
If you do, hmm. What version of Mozilla. One of the early versions had a silly bug / misfeature that it would start up, but if you weren't online it would go into a sort of half-hearted offline mode. Shutdown moz, go online, restart moz. Either or get moz 1.2.1 that seems pretty good.