Hrm... for me, my first ever console game was Super Mario Bros. on the Nintendo. I remember my brothers and I got it for Christmas. We got a Nintendo console bundled with Super Mario Bros. and Duck Hunt (along with the Nintendo Zapper). The reason I say Super Mario Bros. was my first game is because that was the first game I played out of the two and it was the one I played the most. I must have been about six years old at the time.
From then on out we didn't get any more Nintendo games but my parents would take me and my brothers to the video shop once a week and we'd hire out a new game. We hired out new games all the time. Super Mario Bros. 2, Super Mario Bros. 3, Maniac Mansion, Duck Tales, the Flinstones, Crash Dummies, Double Dragons 1 + 2 + 3, Alien 3, Predator, Ninja Gaiden, Wizards and Warriors 3.. the list goes on.
My first step into online gaming however would have been Star Craft, which I started playing when I was ten years old. We went to America on a holiday for a month, and while there my oldest brother bought Star Craft for PC. When I played it online, I tended to only play custom games, like "RPG" maps and stuff.
but like DBucks8, I think what really got me into online gaming and perhaps, gaming as a whole, was also Blizzard's classic.. Diablo. I played Diablo offline first in single player, and upon finishing it I took it to the next level by playing online. I can still remember my character, "Krackshot", a level 37 rogue.