I got into anime by watching Adult Swim when I went to college in 2003.
Well yeah I watched "Pokemon" before that, but because I got the video game and thought it was a good RPG, and "hey its a show for the game".....problem was there were 150 monsters and after they made 150 episodes, they ran out of ideas....
But really, I started watching Adult Swim Fall 2003 because I was up late buried in homework, and Futurama was in rerun (it wasn't on DVD yet)
and Sealab and such was on, but after that...well I watched a few episodes of Trigun, Cowboy Bebop, but never could watch from episode 1, so I didn't keep watching (I have a thing where I know I need to watch all the episodes to know what is going on)
But the first one that really grabbed me was...."Blue Gender". It was sort of like "Starship Troopers: The Series" but most apocalyptic; they *didn't* save the world, the human survivors fled to space stations and the giant bugs killed most of the people on the ground, and there's just this nihilistic post-apocalyptic vibe to it that I simply didn't see in normal TV: I mean *literally*, the two main characters are the ONLY people who live for 3 episodes at a stretch. Everyone they meet eventually dies.
However at the same time I watched some Big O episodes out of order (luckily the first season is mostly standalone), then they started over from episode 1.
So a mix of "Big O" and "Blue Gender"
Then again....I'm not a gigantic Evangelion fan, but by *randomly flicking channels* I watched the first episode during "Giant Robot Week" in February 2003. I had no idea what it was or what it was called, but when Adult Swim then chose to air the full series (without the massive edits) in October 2005, I made sure to watch obsessively.
***Also, Ghost in the Shell: The Series. I mean this was as close to a real adaptation of William Gibson I've ever seen. For a while there, they had a "Ghost in the Shell/Big O" hour from 1 AM to 2 AM on WEEKDAYS. What a time that was.
As for "movies".....I saw "Princess Mononoke" with a friend in early 2002. Later in 2003 at my high school anime club they showed "Akira" and "Ghost in the Shell"
My college didn't have an anime club officially, just some kids from the crafts house who squatted in an unused room and showed a whole wide range of stuff. Really opened me up to a lot of things. Also this is how "word of mouth" I first saw episode 1 of Elfen Lied.