I don't think there's a clear series that got me interested in anime. It's more like I was interested in animation series that were not overly silly.
For example I watched Little Women, Heidi, Captain Tsubasa (was very eager to follow this one).
Other animes came later to the TV channels, but where I live they started to show some more animes like Sailor Moon, Pokemon, Dragon Ball, Ranma 1/2, Hellsing, Cowboy Bebop, Escaflowne, ... and I watched a lot of them. A bit later I also got in contact with Manga (mainly through a magazine that translated the Shonen Jump stories).
So in the end I never really needed that one series to get me interested in Anime - I just generally was appreciating animation series as kid.
Nowadays I don't watch a lot of Anime, but I can still appreciate the good ones when I stumble upon them (Brotherhood and Hunter X Hunter as the imo best Shonen series, Ergo Proxy for its dark, thoughtful tone and sound, Nodame Cantabile for feeling good + nice music, Paradise Kiss for style and a bit more realistic handling of relationships).
I am currently watching the new Hunter X Hunter series and a Dota pro player recommended "Legend of the Galactic Heroes" which I put on halt after 10-15 episodes, so I might come back to this again later.