In the summertime you can take the Metro directly to the beach. And if you're over the crowded white trash tanning and blanket sex scene, enter the hidden away mens or womens area located in the big wooden beach bathhouse. Here you can lie naked and enjoy the sun on your private parts while reading a novel or surfing your iPad. Expect lots of gays and lesbos which can be a curse or a party depending on who you are as a person.
Ordrupgaard is a beautiful museum with lots of interesting exhibitions. Right next to the museum Finn jul's house is located, a famous danish designer and architect.
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.