I got into Anime with Ghost in the Shell (the first movie), and although it might not be the best starting point in terms of accessiblity, it was certainly great to start a few-year-long Anime Fan/Japanophile Streak for me. It has a great visual style, is VERY deep and keeps you engaged even in long, talky scenes.
As Avayu, i'd certainly recommend Ghost in the Shell: Stand Alone Complex, wich is not only one of the best Anime TV Series, but also a very accessible one due to being very "western" in comparison. It's kinda like a police procedual with Cyborgs and Cyberspace.
Cowboy Bebop is also a great recommendation. It alternates very well between funny, serious, adrenaline fuled and sometimes even scary.
Noire is also a very cool although not overly complex series that mainly shines through excellent action scenes and a highly unpredictable storyline.
Other than that:
Animatrix (Anime Shorts based on the Matrix, excellent Showcase in terms of variety and style)
Fullmetal Alchemist (another one of the very best, although i'd recommend the "original" Anime over Brotherhood, wich is more rushed and more "all over the place")
Princess Mononoke (just great)
Tokyo Godfathers (another very accessible, "western" kind of movie that still has a lot of heart)
Summer Wars/The Girl Who Leapt through Time (both by the same guy, excellent visuals, likable characters, fun stories that have also some depth. Summer Wars is a little more complicated and weird though)
And ALWAYS Remember, as with everything, 90% of all Anime is shit. But there is still much good stuff.