Let's see if I can remember this correctly. In 1996 I was a Junior and then Senior in high-school. These were some of my activities...
1) Heavy IRC usage.
2) Designing webpages for my high-school.
3) Writing versions of Minesweeper and Life as Java applets.
4) Commenting on Robert Jordan novels in AOL message boards and usenet.
5) Doing not-quite-legal activities that would have involved AOL not being happy with me if they found out. (think statute of limitations have passed)
6) Playing Federation on AOL.
7) Playing Bolo on school network, and wishing my home connection was fast enough to play at home.
8) Downloading music and games from FirstClass or HotLine BBS systems (before I gained a piracy-conscience).
9) I think I played a lot of World of Warcraft 2 online in the summer of 1996 while using Harvard's fast connection.
10) Trying to figure out what the point of Gopher was, and eventually giving up.
The main thing I remember is that while the Web and email were important, they were both a much smaller portion of my online usage than they are today. I think the turning point was 1997, where the web took over in terms of content.
One amusing anecdote from 1996. I remember overhearing two people who couldn't figure out how to email each other. They decided it was because one of them used Netscape and the other used Internet Explorer, and decided just to use the phone from then on instead.