Showing my age here, but the first anime I remember seeing were the old syndicated dubs of Dragon Ball, DBZ and Sailor Moon that ran on some local stations. I remember looking at Sailor Moon in particular and being like "This drawing style is really weird, why are their eyes so big?"
In terms of stuff I actually got attached to from the word go, Pokemon.
Mine was Seitokai Yakuindomo, watched with my mother when I was child. I didn't understand the jokes and asked her to explain them to me, I gave her a hard time. Sorry mom.