My very first taste of anime was Pokemon. After that it was Digimon, Cardcaptors, the Tenchi Muyo/Universe/In Tokyo series, and Hamtaro (which I watched religiously during the summer it aired on Cartoon Network).
But I didn't start becoming a serious anime geek until Rurouni Kenshin started airing on Toonami and I became obsessed with it. I also taped episodes of Inuyasha on VHS when it aired on Adult Swim.
Those were really good times for me, because I bonded with my then-best friend over those series. Even though she was NT, she became just as obsessed as I was. We spent entire sleepovers doing nothing but watching them, talking about them, drawing fanart and writing fanfics of them. It sounds so simple, but it was really significant to me.
I wish I could find someone to do that with nowadays, but it's kind of hard to find 20-something year old women who would rather watch anime than hit the bar scene picking up guys, or focusing on college work.