For photography, the central area of Copenhagen around the national library (The Black Diamond) and Amalienborg, Tivoli, and Nyhavn, are good for various architecture. The smaller seaside towns are also very pretty in summer or autumn, for example Hundested (1.5 hours by S-train and Lokalbanen).
However, much depends on the weather. In winter days are short, weather is often foul, and the scenes are heavy on that Wallander atmosphere, so you might not get what you want out of that. The Christmas decorations and street activities can be nice.
I recommend visiting in later summer or early autumn, then Denmark is at its most photogenic.
If you want to photograph the Little Mermaid, bring a longer lens, she's tiny. But you're better off going somewhere else and picking up a postcard in the airport, frankly.