Legend of the Galactic Heroes: The best space opera I've ever experienced. Likely the best period.
Tengen Toppa Gurren Lagann: Balls to the walls crazy awesomeness. It will pump you up and make you into a man. ROW ROW FIGHT THE POWER!
Death Note: Brilliant premise and delivery. It's amazingly entertaining to watch the 2 chessmasters of Light and L duke it out mentally. One of the very best here.
Code Geass: Morally dubious protagonist fights his retarded friend with magic eye powers. Actually really good, I highly recommend it.
Bodacious Space Pirates: Ironically maturely written despite the title and with great characters and emotion. I luv it. The main character is so likable and charismatic.
Fullmetal Alchemist (and Brotherhood): You'll love it. It's like a smarter, better told Naruto. Right up your alley. Deservedly one of the most popular animes of all time.
Neon Genesis Evangelion: Amazing. It revolutionized anime for a reason.
Ghost in the Shell: Stand Alone Complex: Very mature and intelligent with great thematic weight etc. Also a popular anime.
Azumanga Daioh: ****ing. Hilarious.
Cowboy Bebop: People love it. It's ok imo.
Hellsing Ultimate: Utterly awesome and dripping in gore and style. Class.
Baccano: Hard to understand at first by seeeeeeriously entertaining. Like, mesmerising. It has dozens of named characters and they are all memorable and badass. Great anime.
Trigun: A classic. The second act hit you like a punch to the gut and has so much weight. Just great.
Kino's Journey: Somber and beautiful. Very imaginative storytelling and deeply fascinating imo. Def. check it out.
Mahou Shojo Madoka Magica: Don't let the magical girls scare you off. This is an amazing series that will blow your mind and make you shit your pants.
Samurai Champloo: Like Bebop, its very cool and well liked.
Darker than BLACK: Outstandingly dark and entertaining. I seriously can't wait until the next season because I fell in love with this one.
Black Lagoon: Deliciously dark and cool. Again, very popular and again for god reason. Check it out.
Durarara: It's brilliant and imaginative and very well delivered. It's just right for your age imo.
Shingo: Veeeeeeeery laid back but very charming and nice to watch. If you're ok without much action it's a good watch.
Revolutionary Girl Utena: A personal favourite. Full of symbolism and lesbians.
Strawberry Panic: Speaking of.... Very girly. But with lesbians. Lesbians make everything better. Les~. Bians~!
Wow, that list is a bit longer than I thought it would be. Brb, sorry for the read.
DBZ and Sailormoon 