Ninja Scroll, Ghost in the Shell, Akira, Princess Mononoke, and Spirited Away were the first anime DVDs I ever bought, in that order.
Prior to that my sole experience was Dragonball Z, Gundam Wing and (the excellent) Tenchi Muyo.
Then I started meeting other people interested in anime, and discovered kazaa and forums, so my exposure increased. Ninja Scroll TV (shit), Gundam Seed (mediocre), Chobits (enjoyable), Excel Saga (funny), Ramna 1/2 (dated), Inuyasha (Dragonball Z for girls), Cowboy Bebop (best show eveeeeeer) and .hack//SIGN (loved it at the time, probably wouldn't now).
The MVC I was working at then got the Escaflowne complete boxset in and I bought that. Fantastic series, though you could really tell they ran out of space to tell the story when the planned amount of episodes got slashed.
And that was the beginning of my anime saga. University broadened my horizons further still, but that'd be too many things to list, and a lot of the passion had gone by then. My fondest memories are still in the discovery of something new - those first five anime DVDs I bought, the first time I discovered P2P downloads and forums telling me what to download, etc.
Spoke to someone at high school who was also into anime and he gave me a shot of Excel Saga on DVD...Jesus, it hurts me to think that I'm never going to get back those 2+ hours of my life.
Excel Saga is brilliant, but
only if you've seen a broad range of anime beforehand. Every episode is a parody of a different genre and tropeset.
Unfortunately, they run out of ideas about 12 episodes in, and the series quickly devolves in quality after that. They then get very meta and the characters become self aware that certain events are happening solely to 'prop' up the series as the creators run out of ideas, which was cute but didn't save it.