Mine was Robotech which I recall blew me away by having such a big continuous story with each ep continuing right from the previous, and I was already into sci-fi, which I think was followed up by a double punch of stuff like Akira and Vampire Hunter D and then I kept renting more and more from Blockbuster like Project A-Ko, Slayers, Ranma, and Kishin Corps and was in for good getting my parents to buy more as gifts.
I did also randomly catch some repeat Speed Racer before knowing what anime was too and don't think I knew what anime was for sure until I wanted to learn more about the stuff I'd just started renting.