This is pretty weird, since I didn't even realize it before much later, but my first anime was actually an anime adaptation of Tove Jansson's "Mummi" series (that's "Moomin" for you Americans) from 1990. What's even weirder is that I remembered the stock sound effects so well that recognizing them is how discovered it was an anime in the first place.
In retrospect, that show was awesome. I swear, if you sing the first few bars of the intro to any Norwegian my age, they'll probably be able to continue the verse. Tove Jansson must have been pretty hardcore too, being gay and born in 1914.
[sub]For a newer example, I started watching anime again when I was 14, with Azumanga Daioh.[/sub]