my main suggestions are pretty much always the same
elfen lied: a series that comes close to perfect, but denies you the closure that you oh so desperately want
Solty rei: a perfect series with a range of emotions and an amazing story line
Vandread: another perfect series that grabs your attention with close observations of human interaction
Claymore: From what i have seen, it is a great series for the beginning anime watcher. It contains a lot of the things that people come to anime to see. Good looking girls, unbelievable twists, artistic views, a range of emotions, monsters, and great characters.
Bokusatsu Tenshi Doku-Dokurochan: a funny, perverted, somewhat adorable series, that is great for people who are into the comedy side of anime.
Basilisk: Like Ninja Scroll meets Romeo and Juliet. This series is one of the best i have seen, having that range of emotions that everyone loves, mindless killers, hilarious characters, and love.
Berserk: A completely human series that goes haywire. War, Violence, Love, and Giant Swords. This anime has enough packed into 24 episodes to please almost any anime lover.
Rozen Maiden: haven't watched enough of it to clearly judge, but what i have seen has been amazing. a psychological thriller, involving dolls that fly around in suit cases. a very unique and interesting series.
also, i recently started watching baccano, and what i have seen was pretty good. not my favorite, but its definitely a good attention grabber.