As long as you stick to using only two or three internet applications at a time, surfing the net on the A600 is no problem.
I have Grapevine set up for IRC and MSN (through Bitlbee), and AmiMUD for Multi-User Dungeons/Domains. MUDs are like a cross between IRC (Internet Relay Chat) and a Text Adventure, with RPG elements. I was very surprised to find so many MUDs in use around the world. If you can't get into the gameplay right away, there are always other users online to chat with who can help you.
The A600 can play MODs as well as MUDs
I use QuadraPlayer for playing MODs on it, which you can see an old (pre-A603) demo of it runing back when it had 1MB RAM here -
[ Show youtube player ]