Lessee… about '85 or so, my roommate and I paid to have the first private phone line installed in our dorm so we could use his Commodore 64 to access the university’s net. We played a lot of a text-based Star Trek game with some other geeks on campus.
But the actual internet? Probably around '95, in Seoul, S. Korea. Mostly e-mail at first but rapidly evolving into full-fledged surfing/wasting time.
My first adventures outside the box probably began in 1988, long before the internet had become a series of tubes. Assuming things outside the internet as we know today still count, I was cruising bulletin board systems, dialing the local library, and checking Usenet by the time I was just eight years old. As soon as I was done, I’d go play Sopwith, Think Quick, or Where In The World Is Carmen San Diego. That’s right, the original one. In CGA. I am hardcore. 
My dad wasn’t really on the cutting edge of tech, but he did adopt modems for use in his business long before they became commonplace with consumers in the early 90’s, so I had a bit of a head start. My memories of watching my parents dial up other computers with a phone was fascinating to me. I grew up around it. As the web slowly came into being, I made the transition around 1993 to AOL, and then eventually to a local dialup provider in 1996, then DSL in 1997. It’s hard to believe I was only 17 then.
Quantum Link (aka Q-Link) in '86/'87; AOL in '94. I got my first email address around '91, it was a feature on some local BBS and was quite long. Never figured out how to use that feature.
Edit: 39 here.
I first learned of and started using the internet in 1992 when I began college. It probably helped that my best friend from high school had gone to MIT and was able to tell me all about it and the wonders of free long distance typing to each other! through the wonders of MUDDing/MUSHing. But, of course, I had experienced the pre-internet in the form of the Compuserve dial-in bulletin board on my Commodore-64, years before that in 1987 or so. College intro’d me MUSHing, email, listserves, gopher, and the very beginning of the web (and cyber-romance!). After graduating was real web, irc, IMing in the form of icq, porn that wasn’t ascii or 2kb gif. My first DSL! The rest is history… But it was all a smooth transition, there was no one moment where it became mundane for me. If anything, what made it mundane was when people in the family further back in generation started using it (grandparents, mom…). That, or when CSI had an episode about the Furry scene.
Oh wait I’m sorry, was the correct answer…
When the Straight Dope was born?
1996 while in High school. Only Juniors and higher could use the internet at that time. They were “borrowing” the connection from the university and when there were to many people on it, it would slow everything down.