Wolf's Rain is fun. The early episodes are a blast even if you're not a big anime fan; by the time it gets a bit deeper in, maybe you'll have acquired the taste. If not, hey, like I said, the first few are a lot of fun on their own, and I think it could pass for pretty good Western animation with a deep-ish story. It's good both subbed and dubbed, which is rare; I think it was on Adult Swim for a while. This is a story-and-action driven anime that does not have big long fight sequences all the time; the few big fight sequences they have are dramatic without being excessive or taking the whole episode. Supernatural stuff, while present, is kept well under control. The music and art are both top-notch for anime. One season, bit a fairly long season.
FLCL (Furi Kuri) is a good way to kill off a Saturday. It's bizarre, has GREAT music that will really turn you on to Japanese rock, has plenty of really deep introspective meaning that you're sure you can figure out if you stare at it long enough, but most of all it's a serious 'wtf' ride. You may not come away liking anime any better, but you'll definitely be weirder for it. Also, epic music. SIX episodes long...so you can get it off Netflix or something, and watch it all on the spot.
Fullmetal Alchemist is one I'm surprised nobody else has mentioned. It's another one that made it onto Adult Swim. It's about equal parts comedy, drama, action, and fight sequences. Hell, I'd say it's broadly representative of good anime--if you want a good summary of what good anime is like, FMA hits square in the middle. Good story, occasionally heartwrenching, minimal but present chibification, magic powers that don't go over the top. It pays enough service to anime conventions that it couldn't be mistaken for western animation, but it's not dominated by them. It's...like, what, 3 seasons?