When I started college, the Internet was pretty much just college to college, and there was no web yet.. that was a few years away. Files? You searched FTP space with Archie. Or glued stuff together with a Netnews/Usenet client.
As far as chat goes, our first chats were on our mainframe, a program called tell, You could message folks, and evel leave messages (though the "answering machine" was lossy, you couldn't assume delivery). In theory tell could work to other Universities over a network called BITNET which existed just a bit pre-TCP/IP Internet. In practice, neither I nor anyone I knew had any contacts at other BITNET UNIs. Then everyone got on IRC. It was a bit odd to see all these non-technical people stare at a computer all day over IRC. I had friends that literally sat next to each other for hours, not turning around to talk to the person 2 feet away from them, but talking on IRC.
I have a soft spot for ICQ. My first contact with the woman now my wife was over ICQ random chat. No romance at first; both of us were dating people at the time. We just BSed for a bit. But once we met face to face, we started dating pretty fast after that.