In comments, Rick Perlstein writes that, "I looked at my first web site (at the public library) in 1996." I don't remember when I looked at my first web site, but I remember what it was. Back when I was a gamer, some video game magazine -- I want to say EGM -- published a little infobox saying that the very first pictures of the new Mortal Kombat were going to be published on the developer's personal web site after press, but you could access them at this address. I had my father take me to the university where he works, as back then, that was the only internet access point we could think of. EGM had misspelled the URL, but we realized it was because they mistyped the developer's name in the address. And so that was my first web page -- some developer's personal site offering sneak pics of a video game.
What was yours?
Update: Best answer so far is Arthur's:
Around 1994, someone called me up to complain about being the victim of a stock fraud. Since representing such folk is my business, I asked him who he wanted to sue. He said he didn't know anyone's name. He explained there was this place called the "internet' where people promoted stocks. The client apparently wanted to sue the internet, or possibly usenet, itself. I found a way to get on line and spent a few days examining the early internet frauds before it became apparent that all the names were fake, and the internet, or usenet, were not susceptible to being sued.
Investors who lost their shirts shortly after going on line for the first time, who wanted to sue the internet (or later, the worldwideweb), contacted me several more times in the 1994-97 or so period.