Welll...
The first anime I watched with full knowledge that it was anime was the Macross Zero OVA, fansubbed on youtube, without sound, in the back room in my grandparents' house. That I still actually liked it is a bit of a miracle. When I got back, I re-watched it with sound, and it was very weird to me, but looking back, it's still very weird.
That, however is not what sold me on anime as a medium.
The first I saw that really got me into anime was The Melancholy of Haruhi Suzumiya, by exposure to tvtropes and a chance encounter with a Dschingis Kahn AMV. TMoS was really cleverly written, well framed and animated, and overall a compelling series, and through tvtropes, I started looking up other anime series.
The rest is history. Well, fairly recent history, anyway.