Some stuff that I would recommend to any newcomer:
Full Metal Alchemist (Brotherhood or original doesn't really matter, but I think the original is a little better for a newcomer)
Outlaw Star
Death Note
Gundam Wing (for some classic stuff)
D.Gray-Man
Yu Yu Hakusho is fun but gets reeaallly episodic towards the middle (near the end of the Dark Tournament saga), so be warned.
ANY Miyazaki film, but my personal favorites are Spirited Away and Howl's Moving Castle
Some titles if you want to dive in a little deeper:
*BECK: Mongolian Chop Squad- Really good if you're into music, particularly rock music. But you really must watch the dub. It's just sooo much better and no terrible Engrish
*Ouran High School Host Club- YES, it has an almost repulsive amount of man-service, but it's a genuinely funny spoof of the Reverse Harem/High School Romance genre. Again, English dub is my favorite on this because Vig Mignona is hysterical, but you can go either way.
*Princess Tutu- A magical girl series about ballerinas... but a REEAAALLLYY good magical girl series about ballerinas. It's basically what would happen if you took an original Grimm Fairy Tale and turned it into a cutesy-looking anime.
*Wolf's Rain- VERY deep, VERY cryptic, almost cold and isolating but incredible visuals and ENTIRELY substance over style. It has some soul-crushing depth, so if you're into that sort of thing... It's really quite fascinating.
Yeah, I like anime.