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.
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.
Just 25 min by train from downtown Copenhagen lies an old and beautiful forest and country area inhabited by wild (but friendly) deer. It's a vast, mysterious space for getting lost in and perfect for spacing out on mushrooms if you bring friends. Stay away on Sundays as it fills up with screaming children and zombie parents. Grab an ice cream on your way back at Bakken, the old David Lynch-ish fairground located at the outskirts of the forest.
A pretty museum located 30 min north from Copenhagen. It has a stunning sea view and beautiful architecture. Here you can explore many exciting exhibitions. They have a great cafe as well.