I remember a a kid I always had a fascination with the whole dragonball series, gundam wing, and ronin warriors. I am aware I sound like a broken record seeing how this fits the criteria of most people on this forum. The whole dragonball/DBZ were just my thing back then although I liked dragonball for the great story lines and character development. Gundam wing and ronin warriors follows suite. So that basically opened the portal to other great and enjoyable shows.
I would say Tenchi Muyo and Cowboy Bebop were the first anime series that really got me serious about watching anime. I remember recording Tenchi on VHS when it came on xD oh wow...good times, good times.
That's what I would do too. If I had to go out somewhere during a new episode of DB I would just pop in the VHS tape and pray that no one would change the channel while I was gone.
my first anime was dragon ball i was one the only kid at school that knew about it at the time since this was before z
I'll be honest I was kind of a hypocrite kid and didn't like anime, however, one day Dragon Ball came on and I was extremely into it and watched like every episode and got into it, but I still didn't admit to myself I liked the style. Then on Toonami I saw Yu Yu Hakusho and Lupin the third and I liked those ones too, but it still didn't make me believe that I liked that style. Finally! I saw One Piece, and even though it was the horribly dubbed 4kids version, I liked the story and the idea so much that I kept watching it, and after thinking about that and the previous anime I decided I should check more out and it turned out I loved a lot of them.
So a little bit of everything got me into anime, haha!
I remember finding Digimon and DBZ around the same time, 2000 or so. Eventually I found Inuyasha and those combined pretty much had me hooked. My tastes in anime are more refined now, but I'm still a huge fan.
This first anime I ever watched was Speed Racer. Then after that came Pokemon, then Digimon, and then Yugioh. And as much as I love all four of those show, they never got me into anime itself (hell, I don't even think I knew that those style of cartoons were even called anime at the time. But I think I pretty much clued in when Yugioh came around). It was when I first saw Inuyasha back in 2000, that I fell so in love with the show, that show was what solely got me into anime. After seeing Inuyasha, I starting checking out more anime, and it was the stories anime's told that I loved most about that genre (granted not all of them were the greatest).
Voltron (yes it was an anime)
The anime series that get me into watching anime was Pokemon and Yu-Gi-Oh. A little later after watching these two I Then started to watch Toonami. Toonami got me watching all these differint kind of anime that I never seen before that I still love watching today like DBZ, Gundam Wing, Yu Yu Hakusho (etc) 
Gundam Wing and DBZ were what got me started. Still going strong as an anime fan for 12 years!