you don't need a 060 to get online, but you'll need at least a ram board to be able to run the TCPIP stack and some web programs simultaneously. FTP and that kind of stuff can be done on 020. If you want to surf with some kind of browser at least a 030 will do good.
you'll need a pcmcia ethernet card. look for compatible cards at amibay or ebay. you'll also need a hdd, but you can also set yourself a CF-IDE + CF card for about the same price which will give you a better performance minus the noise and heat.