The first one I remember that really got me started into watching anime was Vampire Hunter D. But my very first anime was probably Voltron (Golion in Japan).
Dragonball on toonami was technically the first, but I started really watching anime when I caught Love Hina on a random winamp stream at 9pm and was glued to that monitor until the sun rose around 7 or 8am the next day. Hooked instantly.