I use amiga explorer (has nothing to do with internet explorer) to transfer files between my amiga and PC. Or I use my compact flash adapter. The first option is PC only I beleive. But you could get an USB Compact flash reader/writer and an PCMCIA compact flash adapter. With the right drivers I think that would enable you to transfer files from your mac to your amiga.
As for browsing with an amiga.
you'll need at least some extra fastram. Then you put one of those so called tcp stacks on your amiga (miami for instance) and install a browser (Ibrowse for instance).
But you face the chicken and egg problem. How you are going to get the tcp stack and browser to your amiga?
That is the reason I went the compact flash route. You can install the required drivers on a 720k dosdisk and use crossdos to get them to the amiga. Problem solved; you then have a compact flash enabled amiga and are able to transfer larger files.
Hope I havent spoiled the puzzling fun for you but this is a possible approach.
Edit: Check this thread http://eab.abime.net/showthread.php?t=17879