Well, as a child I watched Ghibli films a lot at my aunt's house, but I didn't really associate them with anime like I do now. Later, as a slightly older child I watched a good bit on Toonami and Adult Swim. Nothing complete, just bits and pieces. Naruto (~first season), Bleach (~first season. Loved it and I want to actually watch it sometime), Inuyasha, Shin Chan, some Pokemon, and Afro Samurai (that one might have been on scifi's Animonday). But really it all started years later in tenth grade when I walked over to a friend's house and he was watching episode 98 of Gintama. The Yagyu arc, toilet paper episode. He showed me Death Note, and then I watched Gintama after that (took me 2-3 years to finish Gintama because I slow roll things I like reallllly hard). Gintama was the first anime I watched subbed and it really felt like the first thing that was definitely anime, partly because I was old enough to know and partly because of the language and the content. Looking back it all feels good to me.
Oh and I dont know when, but I have a memory of watching Gunslinger Girl with my brother. I love Gunslinger Girl (I have the manga in my room, I'm going to read it sometime!)