I assume with 'online' you mean 'internet' instead of a BBS...
In that case, you need both extra hard- and software:
Hardware:
- either an external modem or a PCMCIA networkcard. The latter is, ofcourse, much faster;
- harddisk;
- faster cpu (the faster the better);
- extra ram (16 MB or more).
Software:
- TCP/IP stack;
- webbrowser & stuff.
The software can be bought seperately, but it also comes with OS3.9.
Browsing is quite a burden on your CPU, especially when there are graphics on the webpage you are watching/loading. On my 68030 @ 50 with 64 MB fastram, browsing is hardly usable. With a 68060, it should be ok.
E-mail, FTP and chatting doesn't need a cpu like that, as it requires much less resources.