Bleach. It's amusing because it feels like I just got into anime recently with that response, but it's actually been closer to a decade since it started airing dubbed.
Of course I don't follow it anymore, but I do have fond memories of some of the earlier parts of the series when, in my opinion, it was actually somewhat good. The first OP in particular is very nostalgic to me.
Toonami generation here: Gundam Wing, DBZ, Tenchi Muyo!, Sailor Moon, and everything else that was airing in the Golden Age. They didn't all age well, but they changed my life, for sure.
The first anime that I watched on (on my own accord) was Deathnote. You could say your "first" anime was dragon ball Z or something that aired on adult swim 7 years ago, but I'd like to think that your first realistic anime experience would be watching 240p on a broken ad-infested video player on some random site you found when you googled "[insert anime] episode 1 HQ." Oh the irony.
I think my first anime was Slayers on VHS with my cousins, only reason they chose to watch it was because it had a girl on the cover. (IMG:[invalid] style_emoticons/default/laugh.gif)
The first anime I'm consciously aware of watching (though it was called Japanimation back then) is Digimon. I saw others before it, but I can't remember what they were now.
The first anime I watched was probably Doraemon on VHS ... dubbed in Cantonese. At that time I seriously thought the stuff I was watching were Chinese cartoons until I found out years later.
The first anime I remember seeing was Ikkyu-san when I was very little, probably only a few years old. Even now, I can remember the opening to it very clearly, and the teruterubozu in the ending. Don't remember much else, just those two songs engraved in my mind.