While my first anime was probably either pokemon or dragonball it was definitely death note that got me to start paying more attention to the medium and is the reason I watch anime today.
My first anime was Saint seiya ,it was so horribly subbed but i liked the shiny armor xD,and the first i loved was Evangelion even trought i didn't understand the story at all (i was like 8yo ).
Well, if we disregard stuff like Pokemanz and Captain Tsubasa, then I guess the first anime that I knowingly watched was Fushigi Yuugi, followed by Last Exile.