Right so if you're coming to CPH you really need to checkout the following:
Tivoli
Rundetårn (Round Tower)
Frederik's Church
https://en.wikipedia.org/wiki/Frederik%27s_Church
The Little Mermaid
https://en.wikipedia.org/wiki/The_Little_Mermaid_(statue)
Freetown Christiania (Little dodgie but it's a place you have to go see)
https://en.wikipedia.org/wiki/Freetown_Christiania
Nyhavn Harbor
https://en.wikipedia.org/wiki/Nyhavn
Amalienborg Palace
https://en.wikipedia.org/wiki/Amalienborg
Church of Our Saviour
https://en.wikipedia.org/wiki/Church_of_Our_Saviour,_Copenhagen
Gorilla (Make sure you book ahead)
http://restaurantgorilla.dk/forside/
Geist (Little pricey but really worth it)
Papiroeen/Paper Island (Some really great street food)
http://www.visitcopenhagen.com/copenhagen/paper-island-gdk916602