Copenhagen can be pretty expensive, but these places all have good food for reasonable prices:
Mother: convivial Italian restaurant serving sourdough pizza. http://mother.dk/
LeLe: simple interior but tasty Vietnamese food (and lots of it). http://lele-nhahang.com/
42Raw: raw concept in a very central location. http://42raw.com/
Good post from Nancy. I haven't been to aamanns so I can't comment. The other two are splendid. (FYI: Fiskebaren is seafood oriented)
I would recommend Umami and Geist in addition to those. Umami serves japanese and is very fashionable (minimalist decor).
For the best views in Copenhagen, try Sticks'n'sushi at Tivoli hotel or Alberto K at the Radisson hotel. The latter is more formal.