Hi' James. I would recommend you rent a bike to get around town. It's very bike friendly, and you can see a lot since Copenhagen isn't that big. Here's a list of things to consider:

Freetown Christiania is a must see. It's a freetown in the really nice area of Chritianshavn.

Take a trip around the canals of Copenhagen with the Canaltours from Nyhavn. Go for the Netto-Boats which is  cheapest (Same route).

In the inner city there's a lot of museums, attractions etc. which you can read all about here: http://www.copenhagencard.com/?ld=0.

But besides from this I would recommend going to some of the areas outside of the innercity e.g. Vesterbro (Go to Istedgade,  the old meatpacking district and Carlsberg) and Nørrebro (Torvehallerne, Elmegade ,Blågårdsgade) which is really cool and special in each way.

Have a good trip and let me know if you have any questions!

You might wanna consider a bike tour around the city.

It's a really cool way to see Copenhagen which is very bike friendly.

Bike Copenhagen with Mike does a really cool tour of Copenhagen which I would recommend.