Here's a shot of the place Ronny talked about (Dyrehaven in Danish):
The little mermaid has no interest whatsoever, I'd advise you to skip it entirely (unless you want the vacation snapshot). Copenhagen is pretty interesting architecturally, with e.g. the Black Diamond library or the new Concert Hall on Nyhavn. The somewhat new neighbourhood of Ørestad (subway to DR-Byen or Islands Brygge) has some brand new buildings that are pretty good-looking (I work in one of them). More traditional stuff is mostly located in the centre, with lots of churches and old nice buildings. A bit less known but definitely great to photograph are the old navy buildings, just to the west of Kastellet (the fort next to which the little mermaid is located): completely yellow buildings with lovely simple lines, see e.g.
here or
there.
Landscapes are a bit tougher to find. The beach at amagerstrand is quite nice, with a great view of the bridge to Sweden (
here). Dyrehaven is also pretty good. For the skyline, the top of the round tower is hard to beat. The lakes located in the centre of the city are usually quite nice, especially after sunset (
here or
there).