This was something originally I had posted on our club forums in response to:

"Something I've been thinking about recently is how do you introduce someone to anime when they are completely unfamiliar with it. My room mate and I have been trying to introduce my girlfriend lately, as we are giant, slobbering nerds and it is the way of our people.

So what do you think? Do you go the historical route, having them watch "classics" such as Evangelion, or Akira? And how do we even define "historical"? Stuff that's been influential, or the things we watched when we were first getting into anime? Do you start them with "simpler" stuff, or plunge right in? Can you appreciate Gurren Lagaan without having seen other action/robot anime?"

Here is my response:

When I'm introducing anime to people, I tend to suggest things that are not "classics". Classics to me, are either incredibly boring, or stuff that is so "out there", it's really difficult to understand, or people just view it as freaky.

I always reccomend anime first and foremost with story lines that can appeal to anybody. Cowboy Bebop is usually first on my list, with follow-ups like Escaflowne, Inuyasha, and Samurai Champloo. Usually stuff with easy to understand plots, likeable characters, and a mix of action, romance, comedy...like I said, stuff that appeals to everybody. High school animes are generally a good way to go, I feel.

I would NEVER reccomend EVA or Akira to someone new to the genre. Both of these have the potential to scar people for life.

Miyazaki movies are ALWAYS a good bet, especially dubbed. You may hate dubs, but to someone new to anime, they at least want something familiar. Miyazaki movies generally have really high-end actors doing the dubs, which helps ease people into the genre. Two of the best, I've found, are Princess Mononoke and Howl's Moving Castle. Like Billy Bob Thornton, Claire Daines and Scully? Mononoke's for you. Like Christian Bale when he's not cussing people out? Try Howl.

Watch the stuff with newbies, so if they have questions, you can answer them. Even if you've seen Mononoke 15 bajillion times. If you leave them in a room by themselves with anime (even good ones) for the first few times, they might feel you're trying to brainwash them and wont pay as much attention as they should. If you tell them it's good, they'll believe you more if you show them by sitting and watching it with them.

After a few series or movies, maybe say "So, we've watched some more basic things. How do you feel about [insert weird sub genre here]?" Or perhaps ask them what genre they'd be interested in seeing more of. Action? Smatter some Gurren Lagaan or FMA in there. Romance? Try Fushigi Yugi or another mild high school romance. Drama? Something easy to digest, perhaps without as much blood as Monster.

I've introduced anime to newbies using Miyazaki, FMA, and Cowboy Bebop. All three have been a success. Basically, all I've said here boils down to 3 important things to remember:
-Start slow; weird plots that even you have trouble understanding by watching a series at least 3 times is a no-go. It's a huge turn-off.
-Watch it with them. They'll feel somewhat isolated if you don't.
-Introduce them to things that are a mix of genres, until you get a specific feel for what they like.

If you've never introduced someone to anime before, I hope some of these tips help.