I don't think Evangelion is a good intro anime in the slightest. It's hardly something most people can even grasp to begin with, let alone anime newbies.
I would go with Inuyasha. I know it's not perfect, but it has a lot of the different genres of anime packed into one. The plot is pretty simple so you wouldn't have to worry about overloading someone on their first time. Also, it's the show that really got me into anime.
Usually something short and sweet to get the hook in. I use gunsmith cats, any ghibli film, or something like Spriggan or Akira. Gunsmith is 3 episodes (should have been longer), Akira and Spriggan are stand alone movies.
I like a lot of anime series intro and I've answered on this thread but I just remembered that Eyeshield 21 intros were really great like "Breakthrough".