Hmm, I think Voltron was the first one I was really into, although I didn't know it was Japanese. I really started getting into anime when a friend started importing DBZ subbed(it used to be such a pain in the ass to get fan subbed anime) although I had already watched some DBZ in English(near the appearance of Ginyu force).
Digimon, Pokemon, and DBZ. The anime that got me hooked was Yu Yu Hakasho, Case Closed, & Lupin the 3rd. Basically I started watching around 1999 or 2000.
The first full series I watched was Ai Yori Aoshi. Before that I'd watched random snippets of some old school stuff. Don't really recall anymore, maybe some Video Girl Ai or Ranma, definitely some Dragon Ball.
Sailor Moon was my first, but back then i thoguht of anime was no different than american cartoons. Dragonball z was the one that started teaching me tho, and i became hooked when i watched FLCL.
:(
My first anime was when I was 3 and my memory of it is rather hazy. I just recently found out it was called "Galaxy Express 666" I honest wish to see it once more so I can have a full memory of this anime. The only memories of that remains is a space-train and a little boy being cuddled by what looks like a Final Fantasy Shiva.
It's a tie between Galaxy Express 999 and Samurai Pizza Cats, though Sailor Moon is what sank its claws in deepest. Lupin the 3rd and Cowboy Bebop sealed the deal.