I almost forgot! Another one of my "first" animes is Yu Yu Hakusho. (Which was called "Ghost Fighter" locally)
And it's really nice that after 10+ years, I'm still watching it on TV without getting tired. XD
And yeah. I guess there's a difference with first anime, and first COMPLETED anime.
For completed anime, mine was Ultra Maniac on TV, when Animax was new in my country.
And Death Note, for my first completed anime watched not on TV.
Ultra Maniac was epic xD, I even own hard copies of it ><,