I think my favorite moment for Ahsoka was her duel with her Vader, specifically because of the dialogue where she promises to avenge Anakin, proclaiming she isn't a Jedi, and then when she tells Vader that she won't leave him. It was so dramatic and heartbreaking, and the voice acting was spectacular.
I think I became a fan of Ahsoka after the episode "Brain Invaders" in Season 2 of TCW. Prior to that episode, I liked Ahsoka but her character never really stuck with me for some reason, and "Brain Invaders" (and, really, that whole Geonosian arc) just made her stick with me. I think it was the whole drama with Barriss and how Ahsoka handled the situation, which I found really compelling and admirable. I also just found her relatable in general.
Between Ahsoka leaving the Jedi Order or her duel with Vader, especially at the end when she says she won't leave him again. Right now, I prefer the latter but I might still be riding high on the hype... I also plan to re-watch TCW soon so I'll have to see if any other major moments stick out.
As for when I became a fan, that was a more gradual thing. I don't think it was until season 3 or 4 that I began to realize I really enjoyed the character.
Something similar, although I also really liked her introduction not only for adding some much needed relief to the drab of the Clone Wars, but also for adding more dimension to Anakin and Obi-Wan. Don't know how I feel about the story coming full circle, but it had to eventually.
A lot of what's already been said are among my favourite moments, but for the sake of originality I'll say the sewers scene in The Jedi Who Knew Too Much, in terms of acting (and animation) quality I think that was Ahsoka at her best.
As for when I became a fan. I remember liking her as a character through TCW, but I only really became a fan when she left the Order. As they say you don't know you miss something until its gone.