There are just too many animes out there with too many target audiences to suggest one be-all, end-all. Like non-Japanese shows, anime is split into different categories, and among those categories, a thousand sub-categories. I'd recommend doing a Google search on whichever genre you prefer (action, romance, comedy, all three, etc), and then typing the word "anime" after that. You'll get plenty of results.
Personally, the show that got me into anime was Naruto (action), when I was around 13. By now, it's probably too long to commit to unless you already know the story, and I don't even watch it anymore (I quit after the main storyline ended).
My current favorite is Higurashi, a bloody, psychological horror anime with a pretty art style and good, flawed characters. Of course, they're all completely beautiful, but that's a given in anime.
I warn you now, though, that there is a ton of anime out there that just isn't all that great, from a critical standpoint. Even Higurashi has one gigantic flaw and is sometimes hated on for its art style and characters, and Death Note (actually quite good) gets extremely over-dramatic at times (I'll take a potato chip . . . AND EAT IT!!).