The first anime I watched fully knowing it was an anime is Vampire Hunter D back in the late 80's. I still have a soft spot for it.
Dragon Ball and Toonami pulled me back in hard. I loved anime but didn't have much access to it until Toonami.
I had watched some anime adaptations (Voltron, etc) when I was younger, but I had no idea what anime was back then, and not sure those would count.