The thing that first got me hooked on anime must have been Trigun. It was the first anime I watched through to completion, and I still have fond memories of staying up and watching it on adult swim.
Do random pokemon episodes on saturday afternoon tv count? If yes, then that one. If no, then it would probably Bleach, which is the first thing I watched "consciously".
I think it was Sailor Moon or Candy Candy. The first ones I can remember are Saint Seiya (Los Caballeros del Zodiaco, MEX) and what really got me into anime was Dragon Ball Z