Contemporary Art Gallery in the center of Copenhagen focusing on mainly international, abstract, minimalistic art. 

A contemporary art exhibition space located right next to Østerport station, they have a nice little visual arts bookshop as well.

Nestled just north of the city, Dyrehaven is a vast, surreal landscape of meadows, castles, and gnarled fairy-tale like forests — Inhabited by a seemingly infinite population of wild deer. Lovely for runs, strolls, woodland adventures, and the like.

Contemporary art museum

Whenever I feel uninspired I visit the (now genericly named) Design Museum, formerly known as Kunstindustrimuseet. Sort of the V&A of Copenhagen. An old museum filled with gorgeus fabrics, weird artifacts and a cozy library filled with books about arts & crafts from around the world. If you're still uninspired after a visit here you're not able to. Don't forget to have lunch in the beautiful center garden surrounding by the overgrown buildings, statues and old trees.