I found this on another Q
Take one of those silly little boat tours. They are actually really great if you want to get an idea of where you would like to dirrect your trip.
I highly recommend going to Rosenborg Slot (castle). It is in Copenhagen and it holds the Royal jewels as well as a fantastic treasury. Plus the castle in itself is really very lovely.
Also, Frederiksborg Slot, in Hilleroed is great fun. There are extensive beautiful baroque gardens there as well as the beautifully restored castle and royal church. If there is good weather it is a great place to spend the day exploring the castle and gardens.
The little mermaid is a must see of course (however, you can also see her during one of those boat tours.)
The national museum is great if you want to have a look at the viking history.
Of course just wandering around the city is an activity in itself!