When I still lived in Edinburgh, many moons ago, I had a good friend who watched a lot of anime. He introduced me to "FL CL", "Inuyasha", and "Kare Kano". I enjoyed them all, especially "Inuyasha", but I didn't take it any further until...
I was lucky enough to see this in the cinema. Oh my gosh. The art work! The story line! The food! The characters! Suddenly, I wanted to go and find out more about Japanese culture, especially their mythology because "Spirited Away" was just. that. AWESOME. If you haven't seen this film, you need to. I have it, and I'd be VERY happy to play it at club for you. :D
However, an honourable mention must go to "Grave of the Fireflies". I saw this at an anime festival in Edinburgh, not long after "Spirited Away". It was part of a triple bill, which included "Princess Mononoke", "Kiki's Delivery Service" and "Grave of the Fireflies". Why does it deserve an honourable mention? Because it is the only anime, Japanese OR Western, to make me cry.
So, which anime was your first?