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.
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.
My favorite gallery in Copenhagen, located in the meatpacking area... its great for spotting up and coming talents but also established artists from around the world.
Distortion Festival is a 5 days big party, rolling through one neighbourhood each day during the first week of june. In the daytime you'll find free street parties and at night the parties continue in the night clubs with international DJ line-ups.