Shows that are distinctly not stereotypical anime.
Full Metal Alchemist (Brotherhood, too.)
Trigun
Black Lagoon
Outlaw Star (skip the fan service episode near the end)
Cowboy Beebop
Big O
Just about everything with Miyazaki's name attached to it.
Most of these have good English dubbing, sometimes even beating the Japanese dubbing. I'd watch Black Lagoon with the English subs because the dub makes a lot of scenes awkward and confusing (IE: Ok, so we have a character that is supposed to be a Chinese / American who now lives in South Asia, but they always just speak in Japanese to keep things not confusing, and then suddenly they have a job in Japan so suddenly she cuts over to Engrish. So when you have the Japanese dub having that language confusion as a plot device, and then the English dub just axing it altogether, you can see where things get confusing...)
Shows that are decidedly Japanese, but still make good starter shows-
Foolie Coolie (FLCL)- Fast, fun, and zany but it actually rewards you with a deeper story than its less-than-ten episodes would suggest if you're willing to look for it. Even if you end up not liking it at least it's incredibly short.
Gurren Lagan- It's on the long side, but it's still good. Fun to watch, and once you get past the first few episodes it picks up the pace.
Ghost in the Shell- not so much in the artistic style, but Japanese in the sense that they decide to have discussions of philosophy on the side. The movies tend to be a bit more brainy (I feel asleep watching the second one the first time around, second time around it just gave me a massive migraine) than the series (Stand Alone Complex) though don't worry if you don't watch the movies, or the series. Both are set up so that you don't really need to understand the other to grasp what's going on.
Hellsing Ultimate- Don't watch the original series, it's only so - so. Still, it's stylish, and makes for a strong starter for the gory anime. Oh, and Nazis, bad ass non-sparkly vampires, and vampire nazis. How can you loose?
Eureka 7 / Xam'd: The Lost Memories- because both shows follow similar formulas (and were made by the same studio for that matter) I put them both together. Personally I like Xam'd a lot more simply because it doesn't have these annoying mood swings as the series progresses.
Fate / Stay Night- very popular visual novel turned series.
EDIT:
ADVANCED VIEWING: Not for the light of heart. These are the shows that are more famous (or, rather, notorious) for their content.
Neon Genesis Evangellion- Credited for plunging the mecha genre as a whole into a brooding, depressive slumber that it didn't wake up from for about a decade. If you watch it be prepared to watch the series, and then the movies. You don't need to watch the recap movie (Yes, they did a recap movie) though the rest of it actually resolves the series with a non-shitty ending. Mecha genre series that uses three adolescent children (13-year olds) to explore the human condition. Oh, and don't binge on it. Most people who watch it like that report brooding depression after watching it too quickly, and I'll confirm that after watching it in just two sittings I was out of whack for about two weeks.
Ergo Proxy- Either deep, or lol deep depending on how you want to approach it. I couldn't really wrap my head around it either way, and every time it got close to getting somewhere it just lost it for me.
Elfen Leid- Another lol deep show. Lot of potential piddled away by poor directing. Manga mocked the deep aspect with fan service and a laughable plot line. Anime decided that it needed to be deep and fuck the light hearted part. Ends up being a train wreck, but a lot of people still liked it for some reason.