Good luck browsing the web on an unexpanded A600 or A1200! I have 4MB Fast RAM on my A1200 and I still need to turn image loading off to browse the web or I run out of memory. I imagine you'd barely be able to load a browser in 2MB Chip RAM, at least not IBrowse or AWeb (forget OWB, it is only good for 060/PPC/PC).
You really need more RAM, or better yet, an accelerator with heaps more RAM to enjoy browsing the web and going online on an Amiga.
It IS enjoyable though, once you've got everything set up right! SabreMSN, WookieChat, AmiGift, AmiNetRadio and even IBrowse (though it's old now) rock!