I had limited access to the internet by 1993, though it was only through a portal on a local BBS. I didn't get seriously exposed until 1995, when I went to the university. Once I saw what it had to offer, I made sure I had it from them on (and haven't gone more than a few days without some type of internet ever since). :)

Truth be told, though, I didn't play Doom until much later. I can't quite remember the first time I played it, but I'm guessing it was probably Doom II or one of the other versions. I do remember seeing people playing it on LANs (they'd lug their DESKTOPS to their buddies' house to play), but I was out of the loop. One of the guys was gracious enough to let me play for a few minutes, but naturally didn't have the controls down and got beaten quickly and cheaply.

Later on, I did play lots of Mechwarrior II with friends in a lab on campus. I was the lab tutor or whatever, and I was able to talk the manager into letting me hang out there with my friends on some Friday evenings. It was amazing fun. Many years later I picked up Unreal Tournament (2004, I think) and played that a bit on a local LAN, but of course by then the internet was the way to do these things.

The first first-person, real-time game I really, heavily got into was Might and Magic VI: The Mandate of Heaven. I played all the way through that and loved every minute of it. I picked up the sequels and played them as well. I also enjoyed the Elder Scrolls games, though nothing as really hit me with the same charming style of M&M VI.