Hm, I think the first anime I watched completely was the old Sailor Moon series. My girlfriend loved it and I watched it with her and somehow I liked it. Then I became an anime fan.
I also watched Pokémon or Yu-gi-oh that time, as well as One Piece and Naruto, but not regulary, so yes, I would say Sailor Moon was the anime that pushed me completely in the anime world.
What about you, guys ?