Pentium Iii: Good For Internet?

The RAM upgrade does not really affect the browsing speed as much as the speed of your internet's ISP. The RAM upgrade only boosts the performance of programs on your computer (like MS Word, Outlook, etc), including how fast they launch, and so on.
One way your browsing can be slow is when you visit websites that include a lot of advertising banners and/or popup ads. To avoid all that, and enjoy faster browsing, download Firefox from www.mozilla.org, which gives you the option to block popups. You can then install an extension called "AdBlock Plus" which blocks most banners and such, so that webpages load faster.
Hope this helps.

Yes I forgot that Firefox is a memory hog indeed. I'd still rather use firefox on a slow PC and block all those banner ads, than use Internet Explorer and endure all those flashy, annoying banners.
Actually, I believe that increasing the RAM only makes the browser load (and shut down) faster, but doesn't improve the actual internet speed.