The daycare I was at as a kid always played Dragon Ball and Sailor Moon, but their giant eyes and heads scared me when I was little LOL.
I think the first anime I watched that I liked was InuYasha- the first episode I saw was when Rin and Sesshoumaru first meet, how could I not be hooked after that? I remember thinking it would be so fun to just walk around in the forest with someone as strange looking as that
Next was Fruits Basket after reading the manga, then Samurai Champloo a while later. My time was mostly occupied with Avatar: the Last Airbender the moment it came out. I don't really classify it as anime, but think it deserves honorable mention. (Once you get me started on talking about that show, I
won't stop).