As a good parent and Animation fan, I felt it my duty to bring up my children with an appreciation for the artform, including Anime. I think the first anime I started them on was Digimon, but they don't seem to think that counts.
So, if I had to guess, it was likely
Full Metal Alchemist
or
Last Exile

.
What locked them both in forever, however, was
Girls Und Panzer.
But,
funny story, I had a conversation with the elder of my two kids, and he related that the one show I had watched, likely on the Anime Network, was one that informed him, as a young boy, that Anime was quite a bit different from most cartoons was
Sekirei

(I guess I corrupted him inadvertantly.)