My brother showed me the Akira film when I was really young, I absolutely loved the amazing animation and style of it, especially the initial bike scene. The bit with the giant toys scared the shit out of me though. I'm pretty proud that was the very first thing that started it all for me, better than Pokemon or Naruto with most kids these days.
A few years later he showed me Ghost in the Shell, which was more of the same awesomeness. I always loved the style but never really got into anything heavily, I only sort of embarrassedly watched Tenchi Universe and Gundam Wing when it was on Toonami.
It properly started for me when I was looking through an anime wallpaper site and saw pic-related (but without the background texture and text), I thought it was hot, so I watched the trailer for the series on the official website, I thought it was funny so I ordered a box set for the whole series from China. When I got it I watched two episodes a day, each episode twice, rationing it so I didn't blast through it too quickly. It was the first ever series that I owned myself and had the chance to watch from start to finish properly, so I absolutely loved it.
Since then I've completed 57 full series, 18 OVAs and 28 movies.