I'm going to just jump on the same bandwagon half the thread has and add a recommendation for Baccano! Really, with this much volume, if you don't at least check it out I wonder what the point of the thread was.
I was hardcore into anime for a while, so...
I'm also going to recommend the series that got me really started, Kannazuki No Miko (literally the only Yuri series I ever watched, and still my favourite anime) with the disclaimer that it's something you'll either love or hate, and there is no way to gauge which party you will be belong to. I have personally been surprised by how schizophrenic those results are. I shouldn't even like it by all accounts, not with my dourness.
If you really, really like anime as a medium then I would recommend The Melancholy of Haruhi Suzumiya and Lucky Star. But only if you're a lover of the medium. If it was just that there were some anime series you liked and you're curious for more, maybe try something else. If you're entrenched in the sub-culture, they will seem like god's final gift to his creation. If you're on the wall they won't be what makes you a convert. Or maybe they will. If the previous paragraph is any indicator there isn't any way to gauge what brings people in.
If you like mecha but don't like being preached at, then Mobile Suit Gundam 0083: Stardust Memory is a good choice. Odd for a Gundam series to not just be left wing propaganda but I guess that's what I found so refreshing about it, and as an OVA, the animation quality is significantly higher. Do note that it is very heavily front loaded, the home stretch is, for lack of a better term, a clusterfuck. Not an unwatchable clusterfuck, but a clusterfuck nonetheless.
If you aren't one of those people who blindly hate the church, to whom the prospect of a clergy protagonist isn't a cause to foam at the mouth, then Chrono Crusade is something to look into. It's well written and animated and all that good stuff that makes for a good quality anime series. Like 0083, however it's concluding arc lacks sufficient cohesion, and it has a Christmas episode, my mortal enemy.
I'm also going to recommend Black Cat. Before any otakus descend upon me like the thousand nations of the Persian Empire I'm going to note that there is probably more in the mangas, but the anime is pretty good as well. Of course, in a series with magical powers, a villain who tries very hard to be a good Sepiroth homage but only winds up being a bad imitation, and two climaxes, it will be the little details of the general plot that wind up killing your suspension of disbelief the most. And they will, trust me.