My first fighting game is street fighter II. In my opinion, it was actually fun. The fighting game I took seriously, is Soul Calibur 2 for the gamecube. I would spend hours playing as Link, Yoshimitsu, and Xianghua.
My first ones were MK2 for PC and One Must Fall - 2097. The latter featured an extremely well-written storyline, some great characters and a challenging tournament mode. A cult classic, if there was one.