An old Amiga is really good at doing all the things you could do on the net back in the 90s when the web browser was still being developed, but that was long before HTML5. It doesn't have Flash or CSS support either, so most modern websites don't display correctly at all. Amiga.org has a proxy set up by Karlos which makes it totally usable in IBrowse and AWeb (the two main classic Amiga browsers). But apart from the web there are heaps of things you can do just fine, like IRC, MSN, FTP, E-mail, MUDs, P2P filesharing, Streaming Internet Radio, and playing some cool online games like Dynamite, which is an online multiplayer Bomberman clone.
Here's an example of some of this software running on my A1200 with my 030/28Mhz card and Workbench 3.1:
http://www.amiga.org/gallery/images/5087/1_1024x768.pngHere's a video of me playing Dynamite and listening to a few streams through AmiNetRadio, unfortunately YouTube cut the audio out so it kind of ruined the video for me, but this was streaming music online with a 030/50:
http://www.youtube.com/watch?v=j72NjheWP-A