It's very complicated to me to tell which anime is my first one, but I try to explain it:
My first "anime" was:
Top Striker, Sailor Moon, Transformers, DBZ, but that time I didn't know they are "animes", and I prefer them as "cartoons". Also I couldn't remember exactly which is the one that I saw first. So to be honest I didn't prefer these to be my "first".
My first anime that I know it's an anime and not a cartoon:
Record of Lodoss War OVA, Armitage 3 and Gunsmith Cats. I saw them back-to-back so I conder these 3 animes to be my "first".
The earliest I ever seen was when I was VERY YOUNG when my parents bought these movies for me or I saw them as Saturday morning cartoons. Basically, before I even knew that they were other country cartoons.
I watched Naruto cause I thought it was hilarious the college kids (my brothers friends from college) watched anime and that was what they liked, and then it just got me hooked. Kind of one reason why I still cant let go of Naruto, I just love the ninjas. And until there is a better ninja anime, im still not going to say Naruto is shit.
That might be Sailor Moon for me, but then, I just saw some pieces of it when I was zapping on the TV. The first animé that I really watched and liked was Midori no Hibi, that's the one that got me to start watching animé.
I saw that on tv when i was may be 5. Had no idea it was anime back then but i had a really strong attraction to it unlike other cartoons, for some reason. 11 years later when i officially learned what anime is i now know why i liked it so much back then.