First anime I remember watching was Sailor Moon (I was in bed for months with a foot injury, and we didn't have cable TV, or else I wouldn't have watched it) but the first series I watched b/c I got into it was Fushigi Yuugi (also a very 'girly' anime, and I will freely admit that I cried like a baby during some of the eps). That was back, I think, before Cartoon Network even existed.
First movie was Akira, way way long ago. I didn't understand it, but totally loved it. But that was back in...jeez, something like 1995, I think, and I didn't see any more anime (again, that I remember as anime) until Sailor Moon. But when I was growing up, US networks were importing and dubbing some anime for release as cartoons, and I may have seen one or two then.