Monster. It's fuckwin as long as you're not a moron who needs giant animu eyes and Gainax boobie-bounces to watch a series.
Kaiji. Tense motherfucking series. Kaiji is a tsundere motherfucker, too. Akagi is a good choice too - it's actually complete and it's got SEXY MAHJONGG ACTION. Watching the first few episodes of Akagi will probably give you enough to know whether the rest of it is something you'll like.
Hayao Miyazaki movies are good for everyone, always. So, Princess Mononoke, Spirited Away, Nausicaa, and Totoro. Howl's Moving Castle, you can miss that, it's not so hot.
Gurren Lagann is not as good as people like to pretend it is, HOWEVER, taking into account that you're a complete n00b to anime, it's a good starting point for all things mecha and fanservice. More experienced viewers are bored with it, n00bs think it's the holy grail. Watch it instead of Eva; Eva is, even moreso than Gurren Lagann, overrated trash.
Turn A Gundam is definitely the best Gundam series. Watch it after Gurren Lagann to see how a good mecha series is done.
Filler shit for when you're bored: Melancholy of Haruhi Suzumiya, Baccano!, and Mushishi. Various genres to get you started on your way to becoming a true anime fag.