I have two answers for this.
My first animes in which I knew they were anime was Card Captor Sakura (CardCaptors at the time), Pokemon, Digimon, and pretty much the Toonami line up late 1990's - early 2000's
However my actual first in which I had no idea what anime was from ages 4 - 9 were Little Nemo: Adventures in Sumberland
and Puss'n Boots: Around the World in 80 Days/Travels Around The World
with a clip or two of Sailor Moon. I use to have the VHS of both movies and watched them so much Little Nemo was pretty much ruined. Luckily I now have that on DVD but there nothing on the other movie anywhere and I lost the VHS years ago =(
I just recently found out those two are anime. I find it kinda funny now that two of my favorite movies as a kid were indeed anime and I'm such a big anime fan now.