Another vote for Round Tower, that was really good, as mentioned they have some interesting exhibitions on too which are worth a visit.
Bars/food – get a ferry over to Refshaleøen, there there’s a street food place and a Mikkeller bar (Mikkeller Baghaven).
Up towards Nørrebro is a place called Brus – great microbrewery and tasty food.
Around the meatpacking district is good: Mother does decent, reasonably priced pizza, Bodega 54 is a punk rock bar with a strong St Pauli theme, and opposite is a bar called Fermentoren which is good too. Mikkeller Bar Viktoriagade as also a good shout.
Torvehallerne is a covered market place with nice food stalls (some great Smørrebrød)
I found Christiania pretty weird to be honest. As amusing and novel as it was to see copious amounts of weed and hash being sold in the open, I felt a definite seedy undercurrent there.