Two museums that I hit that I would recommend (both free!): The National Museum and the David Collection. No pictures allowed at the David, but it was a really amazing collection of Islamic art that was mesmerizing. It made the rest of the collection of Danish and European artist look amateurish.
The National Museum was huge. Denmark has a policy that all artifacts from the past can not leave the country, so they have a wealth of artifacts reaching back to the stone age. There were well-organized rooms full of creepy peat-bog sacrifices and Viking artifacts.
Awesome Rune stones, and lovely plundered treasures abounded.
Mostly though, I just loved walking around. It's a very pretty city. There is a ton of construction to get around as they are extending the metro, but there were plenty of places to see otherwise.
The lovely old stock exchange:
The vibrant, picturesque and touristific Nyhaven.
It was mobbed, with tourist spilling out of the cafes and people getting go-cups to sit on the pier with.
Amalie Garden, next to the marble church.
You wouldn't have guessed it from all the tourist shops slinging figurines and postcards, but yes, there is a mermaid chilling out in the harbor.
Even though there were signs saying not to climb the statue, plenty of people, ahem, men, were gamely climbing the rocks so they could have their picture taken giving the bronze a nice grope, proving that tourist can be dicks all over the world.
There's no pictures allowed in Christiania, and for good reason. It's an old military base taken over by squatters and made into a community of hippies (or hippie-minded) and tourist coming to buy pot, and quite a few prudes rubbernecking and looking around nervously at the whole thing. It was really unique- there is a "pusher street" filled with hash and pot stands, all covered up with military camouflage (a joke- the government wanted the dealers to be "less visible", so they now hang out under camo canopies).
There are rules posted: Absolutely no pictures, no weapons, no biker gangs, no body armor, no fireworks, no hard drugs, no running (it creates panic), no cars and no selling of stolen goods. Cannabis is not legal in Denmark, but somewhat tolerated in this one area, and they are trying hard to have their idealistic self-governing hippie utopia. It's really a unique place. I'm sure it has the ability to scare some people off, but there was some really unique homemade architecture and art works all over the place and it's totally worth a look even if you aren't buying.