Since we seem to sharing our personal histories with the show....
My own beginning was... unusual. I definitely remember that
Carnival of Monsters was the first serial I watched. But there was a small gap (weeks, not months, let alone years) before I watched the next, and I somehow didn't key in on the fact that, by then, I was watching
Fourth Doctor episodes rather than
Third Doctor. In fact, I didn't actually key in to the fact that there were multiple Doctors until I found the Doctor Who novelization section of my local Waldenbooks, and saw the cover of the novelization of
The Five Doctors. (It was still a little bit before I realized that
Carnival of Monsters had a different Doctor than the one I was then currently watching, but suffice it to say, I figured it out eventually)
Thus, while the Third Doctor is technically my first, I consider the Fourth Doctor to be where I really started watching. That said, I tend to gravitate to the Seventh Doctor as my favorite of the Classic era (perhaps in part due to the fact that his is the incarnation I've tended to cosplay as at conventions the most, although I also do Four and Fourteen, with "mostly there" versions of Eleven and Twelve that I don't really do all that much anymore, having settled on Seven... including a Movie variation as of this past February... and Fourteen for most occasions. Even for Four, I tend only to do him after I've done the others at the same event, and not for as long, because the coat is HEAVY!).