Very first map I played was the one with trains (cp_well, thanks iRC!) and I believe I choose Heavy. I remember because I got my first achievement there (Iron Kurtain? - Take 1000 damage in one life). Though I don't really count that as my first "class", since I only played for 5 minutes tops.
My second map, which I call "First" was 2fort as a Sniper. I had no idea what the other classes were and Sniper seemed pretty reasonable. I played CS:S mostly before and "knew" how to properly play a Sniper. I soon learned to kill someone without a headshot is by charging up, or keeping the gun scoped in. I was so used to CS:S that I would press "qq" to quick-switch after a shot. Didn't work very well. ;P
I call that my first because that was the game where I really got into TF2. I wasn't Sniper for long, as I saw the "Overpowered", 100% aim Sentry Guns and switched to Engineer. All I did was camp by my teleports which I only took to get as many as possible and whack my Sentry. Yeah, I got bored fast and switched to Pyro and was pretty easy to learn. Pyro had been my main class for quite some time, but now I often fluctuate different classes whenever I'm on TF2.