In theory you can surf the net, a friend and I recently did this on my IIci after getting ethernet working. The problem is that most modern websites use java, flash, and other scripting languages that are not supported by old browsers. Even having a bunch of large (by late 90s standards) photos will suck up memory and slow things to a crawl. You can find a few old style websites out there though, or browse the internet archive. Unless you have a fast '040 machine it will be painfully slow though.

It's one of those things you might do just because it can be done, but it's certainly not the most enjoyable experience you can have on a vintage machine.