Hmm, trying to cover the bases here.
Fist of the North Star. Classic post-apocalyptic anime, staple of manliness, and a fairly old shonen. Even some of my casual friends like it which is saying a lot seeing as some don't even touch other anime.
Also, Planetes. It's fairly realistic sci-fi drama in terms of tech, though nothing much happens at first since it's about space garbage collectors; I was more of a fan of the manga but the anime's good too. It's also directed by the guy who would go on to direct Code Geass, if that means anything to you.
FLCL. Weird as hell. It's short but it's a good exercise in symbolism if you want to feel smart. If you don't, there's still the top-notch animation in every scene, action or comedic.
Dead Leaves. Basically a prison escape from a moon colony. It's pure, mindless action; don't even think about the plot, it makes less sense that way. Directed by one of the guys who made FLCL so this whole thing comes together as an acid trip with gunshots. A guy has a fucking drill for a penis, tell me that's not the pinnacle of animation.
Neon Genesis Evangelion. Infamous mecha anime and, really, you'll either love this and it's the holy fucking grail or hate it with all your being, both for its characters and the ending. Practically required watching in some circles of anime fandoms. Honestly, it's less confusing than people make it out to be but it can still be a lot to take on in one sitting. Also, to properly finish watching you need to watch End of Evangelion.
Lupin III. Very old, very long since it ran from the early the seventies to the eighties with some OVAs and movies in between, so you can pick and choose to some extent. Hijinks, thefts, chases, perversion eveyr now and then. There's a new series The Woman Called Fujiko Mine, if you want something shorter and more modern; plus, holy shit that delicious art style.
Revolutionary Girl Utena. It's considered the shoujo counterpart to Evangelion, sometimes called Eva with magical girls, roses, and lesbians, but it doesn't cause as many shitstorms. Symbolism abounds here, bro.
Cromartie High School. Freddie motherfucking Mercury's a character in this. That's all you need to know
Azumanga Daioh. Basically the daily life of schoolgirls, only it's a comedy and there's not much fanservice. I laughed, I cried, I hnnnng'd. Along the same line, you might want to try Lucky Star, or Nichijou, the latter specially for some surreal shit.
In general, most of Studio Ghibli's stuff is well known among casuals, so stuff like Spirited Away or Howl's Moving Castle falls under this banner. There's also Grave of the Fireflies if you ever feel like you could afford to be depressed for a couple days.
Boku no Pico. Rite of passage. Do it.