Copenhagen Meetpacking district is a good place to go out for a beer. Duing the summer the hole place is packed with young people. There is several bars, galleries and restaurants in the area.

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.

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

A Kilometer stone (circa 1925) stands directly outside Nørreport station, in Copenhagen K. It's easy to pass by, but definitely worth noting if you're in the area — The typographic work on it is incredible.

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.