Um...this is gonna sound really odd, but my first anime was probably The Guyver ._.
My parents rented it, figuring it would appeal to my Power Ranger love. I just remember a few disembowlings before my dad ran it back to the video store. Never did see it again, honestly.
After that was definatly pokemon, followed by Digimon, then Yu-Gi-Oh, then Kenshin, and finally Inuyasha, which finally propelled me into singling out anime as an independent genre.
So...yeah.
~Dorian Gray