Probably my first exposure to fighting games was Street Fighter 2 in this arcade/snack bar at a department store near my house. I couldn't play the game for beans, but fsr I was getting all fangirly over M.Bison and Blanka >> I was just like mashing buttons and stuff and died really fast.
The first fighting game I technically PLAYED was Pro Wrestling on NES, this girl in my apartment complex had it and fsr we used to play it all the time, I always picked the green monster =P
The first fighting game I owned was probably um, Clayfighter on SNES? Or um, Fatal Fury 2 on SNES which I don't even know why I ever had.
The first real fighting game I ever properly played was Darkstalkers 3 on PS1, I'm just gonna go with that =P