Then you absolutely need to see
Madoka and
The Melancholy of Haruhi Suzumiya.
Lets, see, I'll try to list some notable anime that came out since 2006.
Fullmetal Alchemist: Brotherhood was a really good show. Especially once you got past the first 15 episodes or so that closely followed the first anime. Much better ending then the first anime. Lots of action and a good plot, more romance then in the first anime.
Cross Game was a big hit at AnimeSuki, won best anime of the year in the Choice Awards for 2010. Adachi's best work since
Touch. You like romance and slice of life so you should check it out.
K-On! is probably the biggest hit in the Slice of Life category lately. Gets a lot of haters in addition to passionate fans. I don't agree with either extreme, but I think it's a fun little show if you like nostalgia about high school or being in a band.
Clannad is another huge hit. First season is a comedy/romance/slice of life.
After Story is heavy drama, rather well done I think.
Lucky Star was popular. It's more of a vehicle for otaku themed jokes, although it does have some slice of life elements. Watch Haruhi before this one though, otherwise you won't get the half the jokes.
For slice of life you should certainly check out
Bunny Drop.
Shakugan No Shana Action. First season was good, second season was crap, third season was good again. (Also, obligatory meme: Shana is HOT!)
Toaru Majutsu no Index Ha... what to say about this show. Action/Harem. Very interesting premise and setting. Choppy execution. Parts are painful to watch, but others (particularly the Sisters Arc) draw you back to rewatch it.
A Certain Scientific Railgun is a spin off following Mikoto's adventures (but the anime didn't animate her side of the Sisters Arc as told in the manga, very frustrating). Railgun has better execution, but I just don't feel the desire to rewatch it much. Personally I like the manga version of Railgun best.
Soul Eater Action show with some plot. Ending was anime only and little weak I thought, but over all a good show.
Chrome Shelled Regios Action/Harem, Decent show, ending was anime only and again a bit weak. Felli Loss was hilarious though.
The World God Only Knows is an affectionate parody of the dating game genre. Great music, and actually pretty fun. Anime didn't get into the heavier plot elements that you can find in the manga, but both are good.
Tiger & Bunny Action, a bit of an ode to Silver Age superheroes.
Otome Yōkai Zakuro Action/Romance
Ano Hi Mita Hana no Namae o Bokutachi wa Mada Shiranai (Also knowm as Ano Hana). Romance and Slice of Life.
Katanagatari Action/Romance
Mirai Nikki Action, the plot has a lot of great twists, but can occasionally have a few holes. Best not to think about it and just enjoy the ride. Also, has the ULTIMATE Yandere. (She's basically the star of the show).
Toradora! A romance and slice of life.
Ben-To Action slapstick. Ruthless fighters meet in a battle royal for the half price Bento. Only the strongest will eat!
Les Misérables: Shōjo Cosette from the masterpiece theater might interest you.
Oh My Goddess had a series come out, remaking the OVAs, and then extending the story.
Angel Beats! Black Comedy/Drama/Tearjerker (yeah it was a strange mixture). Overly compressed, but still interesting.
Kannagi: Crazy Shrine Maidens is a bit of an unwanted harem, with various hijinks.
Kimi ni Todoke Cotton candy sweet romance.
InuYasha finally finished.
Steins;Gate A Sci-Fi Mystery Romance
Spice and Wolf, an anime about economics and trade in Medieval Europe.
Bodacious Space Pirates Highschool girl inherits her father's ship and privateer license. Adventure ensues.
Mysterious Girlfriend X Sensual but chaste romance.
Hayate the Combat Butler Hayate was the worst parents ever. Comedy/Harem.
Code Geass was very popular, especially with Death Note fans. I enjoyed it until a certain incident and then dropped it like a rock.
The new Evangelion Rebuild movies came out. I liked the first one, but didn't care for the second at all.