EBHughsThe1st said:
A Lupin the III movie.
Like Castle of Cagliostro.
Or maybe a Miyazaki flick.
Castle of Cagliostro IS a Miyazaki flick (he directed it.)
OT:
Series suggestions:
Cowboy Bebop is a fine idea and one I heartily support. A few other suggestions:
I got my start in anime with Escaflowne. It's got a little something for everybody: action, drama, giant robots, magic, pondering the human condition. Prior to that, the only anime I'd ever really watched was Pokemon.
Wolf's Rain is very interesting, story-wise and aesthetically, and I think a first-timer could follow it pretty well.
I might tentatively suggest Noein: To Your Other Self. The elementary school drama gets a little silly, but it has a unique art style and mixes in a dose of quantum physics, if you like that sort of thing.
Tengen Toppa Gurren Lagann IS awesome, but I recommend putting it off until you have a firm grasp of certain common anime stereotypes and concepts. A lot of the show explores/takes the piss out of these.
Similarly, the previously-suggested Lain, Evangelion and FLCL, while all fine shows, would probably be better reserved for when you have a bit more tolerance (for want of a better word.)
Movie-wise: Anything by Studio Ghibli, particularly if it's directed by Hayao Miyazaki. Princess Mononoke is my personal favorite, plenty of action and drama there, but Porco Rosso, Pom Poko, Castle in the Sky, and most of the rest of the Ghibli library are all great. Sure, a lot of them are "aimed at a younger audience", but not watching them based on that only does yourself a disservice.
That's my two yen, anyway. Happy viewing!