Download AmiTCP from Aminet and install that. It's your best option for getting online with an A600. Don't bother with Genesis (bloated AmiTCP client) or Miami (still too big for your A600). Which PCMCIA ethernet card are you using? I have a few here that I use with the A1200s and A600s, and there are three different drivers to choose from which work with many different cards.
I think this is the network boot disk that Thomas is referring to -
http://jpv.wmhost.com/NetworkBootDisk/It contains all the drivers, a copy of AmiTCP, FTP client, and a text-based Wizard to help you get it all running and connected to your network. I don't think there is any internet software on there though, so you'll have to download that from Aminet. I'd recommend Grapevine for chatting on IRC, and AmiMUD for chatting and playing online in Multi-User Dungeons (who needs WoW when you have an imagination).
Unfortunately you won't be running a browser with only 2MB RAM. Older versions of IBrowse will run on the 68000, and work on some older or more basic websites, but you'll still need an internal FastRAM expansion to use it, or some PCMCIA SRAM and go online with your Serial port (which won't be much slower since the A600 can't really do anything fast online anyway). For example, browsing the web on my A1200 030 over Ethernet loads websites at about the same speed that it did over a dialup modem, simply because the CPU takes more time to render the pages than it does to load them into RAM.
But for transferring between the Pegasos and the A600, either a Serial Null-modem cable or PCMCIA ethernet cable and that downloadable boot disk should be all you need. There's also the Hombre disk which is useful for transferring from PCs to Amigas -
http://wiki.abime.net/transfer:hombre
