Start the morning with a free breakfast at your home base, Generator Hostel, a stylish budget accommodation with low rates and lots of fun extras like free yoga and a full lending library. Use the computer center to check up on any local events or festivals.
Walk 15 minutes southwest to the Nationalmuseet, Denmark's impressive and enormous National Museum, which is free every day. The artifacts here - housed in a glamorous Rococo palace - would take several days to cover, but for a quick two- to three-hour tour head upstairs to the Inuit art and culture displays, then over to the collection of bog-preserved remains from the Early Bronze Age.
Lunch is at Cafe Sorgenfri, a hole-in-the-wall cafe that caters to locals with hearty, stick-to-your-ribs food like herring and potatoes or juicy roast pork. This location has a particularly good menu of snaps, a Scandinavian liquor that's served ice cold in small glasses. The menu may not have English translations, but the waitstaff will be happy to help you order.
Grab a cheap rental bicycle from a Bycyklen station or at any of the numerous bike shops in town - rates are about 20DKK per hour - and head towards the west side of town for the Norrebro Flea Market, a popular community sale that is famous for its hidden treasures. Toys, furniture, design items, art, and vintage clothing are priced to sell, and most vendors will speak English, so don't be afraid to haggle.
After a bit of shopping, stop off at the nearby Assistens Cemetery, a peaceful green oasis that serves as the final resting place for some of Copenhagen's most illustrious citizens, including Hans Christian Andersen, Niels Bohr, and Soren Kierkegaard. Entrance to the cemetery is free, and the picturesque graves are surrounded by acres of lush green parkland, the perfect spot for an afternoon stroll and a little heavenly contemplation.
Head back towards the center of town to drop off your bike and walk out to Freetown Christiania, Copenhagen's own hippie commune. This area was founded in the 1970s as a tax-free haven for peaceful living, and has survived by engaging in a robust marijuana trade. There are several nice parks here, and a two-hour stroll should allow you to see most of the grounds, including the residents' unique homes - many emblazoned with graffiti murals - and their carnival-like open-air markets.
For dinner try some of the healthy fare at Morgenstedet, an all-organic vegetarian restaurant that embodies the hippie ethos of Christiania. Try the butternut squash and peanut stew, or a salad of coconut, apple, and red cabbage, and eat at communal tables spread out under the trees. Service here is minimal, and in keeping with the community emphasis you'll be expected to clear your own plates.
After dinner it's time for a few drinks at Eiffel Bar, one of Copenhagen's best down-home watering holes. There's a jukebox straight out of the 1950s here, and the crowd is its own unique mix of students, elderly locals, and happy-go-lucky regulars. The beer is cheap and the atmosphere convivial, so don't be afraid to strike up a conversation. Stay as late as you like - it's open most nights until 4 am - and the walk back to your hostel is just fifteen minutes.