So, since we're all saying what we started with ...
I have to say that it was 1961 or 1962. A friend of mine had Tactics, the AH game that came out before Tactics II. The counters were round, not square, but as I remember, the board was pretty much the same as Tactics II. He eventually gave me the game (it cost him $2 new, I think it was), since he never played it. Of course I threw it away when Tactics II came out. I'd like to have that one now! I'll bet some collector would pay more than $2 for it.
I went from there to Gettysburg, Tactics II, D-Day and just every other AH game ever made until about 1972. After 1972 I was a little more selective in the AH titles I bought. (Most of them were crap, by that time.) That's when I got my lifetime subscription to S&T. After that I got just about everything they, and all their successors, ever did until about five years ago. I still do the mag, but have migrated to computers exclusively for my gaming pleasure. I have game titles from many, many publishers, so I have a HUGE wargame collection scattered around the house. Maybe the next hurricane will take it from me so I can claim the insurance.
i think i discovered AH around 1963 or so with their D-Day game, although i went back and got Tactics II later... i'd been playing other "war games" like chess, Stratego, Risk, but the hex-map games were a real eye-opener...