Way back when I was a kid I'd always fall asleep watching cartoon network and one night I just randomly woke up in the middle of the night and saw the end of Robot Carnival and I was wondering wtf I had just seen. Then Vampire Hunter D started and it was the best thing I had ever seen.
First one I deliberately watched both as anime and to follow the story must have been either Blue Submarine or Noir, as those were basically the only two that would air here as freetv at the time (about a decade ago).
But as part of some TV afternoon cartoon program, ie. not marked as anime and just coincidentally? Probably Sailor Moon. For some reason I also seem to vaguely recall that chibi spin-off of Ah! My Goddess.
Pokemon I believe, or DBZ. But the first anime I download was Shakugan no shana. I still can't believe why I was so ignorant about how to get thing on IRC. It seems silly now.
My actual first anime might have been NGE. Parents had the VHS version back then and gave it to me to watch. I was around 10 back then so needless to say I had no idea what the hell was going on. Even was kinda depressed after I watched it, dunno why. Nowadays its one of my favorites though.
Fucking Dragon Ball Z. I never liked it at all. I only got into anime much later when cartoon network started showing things like FLCL and paranoia agent.