You'll more than likely want a PCMCIA Ethernet adapter, more RAM and an 030 accelerator before even considering surfing on an A600. Indivision ECS might come in handy too.
Then you'll need a Browser like iBrowse and a TCP client such as Miami. YAM is a good Mail/Newsreader program.
Depending on what sites you like to visit, the internet experience will more than likely suck though using the above minimum specs. Heck, even on a hi-end legacy system, you'd probably be disappointed. E-mail is fine though.