My first anime that I have ever watched was Trigun (I know, an older one). It was during when I was going with my brother when he was 15 years old to an anime convention and he and his friend let me watch it with them. I couldn't go to the convention, but they were trying to let me understand what anime was so that I can go with them in the future (which I do now when we all aren't too busy). Anyways, so I watched Trigun and I absolutely loved it. At the time, I loved the style and I enjoyed Vash's humor. I even enjoyed trying to find that black cat in every episode of the series when my brother's friend pointed out that it appears in every episode. Today I still like it, but I wish that they could do a season 2 or something as it feels like it needs something more to it!
That's a tough one for me to pin down since my memory of my first anime is a bit blurred given that most of what I was watching at the time was saturday morning cartoons with morning/ afternoon anime blocks from Toonami during the weekdays. If I had to pin it down though I'd say it would be Card Captor Sakura or Cardcaptors as it was called in the U.S. or Sailor Moon. They usually aired before other shows like DBZ and at the time Digimon, Pokemon, and Medabots were only airing on saturday morning cartoon lineups. But everyday after school I would tune into Toonami and watch Cardcaptors, DBZ, and then something else that is escaping my memory.
Though if we're referring to the instance in which we recognized what anime was and have seen called it that, then that would have to be Yu Yu Hakusho or Cowboy Bebop. At the time everything that was animated to me was referred to as a cartoon. It wasn't until I was slightly older roughly 12 or 13 at the time when I recognized the term "anime" and it referred to basically 90% of what I was watching at the time and Yu Yu Hakusho and Cowboy Bebop personified that for me at the time.
Now I'm older and can recognize just about all the old anime i used to watch lol.
I remember when I was really little, and visiting my grandma, she had a bunch of Inuyasha dvds. I don't remember if they were movies or season sets, but I watched them over and over again every time I went there. Began my love for animated shows in general, and once I realized there was more of Inuyasha than what I saw as a kid, I was extremely excited for that.
Captain Harlock. I was like 5 or 6, and i came cross the old anime that was playing in a local TV station. My childhood was never the same. Cartoons would not do any longer after that. Anime became a must watch!