Here is a list of top tips for half day trips outside of COPENHAGEN.
Louisiana museum - most beuatifull museum in Denmark. Located a little bit outside Copenhagen. find out more on their website. http://en.louisiana.dk/
Dragoer (i grew up there) - small town outside of Copenhagen - take the bus there, only if the weather is nice. close to the airport. go there to have an ice cream on the harbour at nam nam. strawl the small streets of the old city. all in all this is a nice thing to do for a 3-4 hour morning exploration if you have had enough of the city.
Bakken this old amusement park is located in the forrest where the king used to hunt! great for an afternoon lunch to!
Vikinge museet i Roskilde if the weather is bad and you want to learn more about the vikings, go to this museum to get more insights into this era of scandinavian history.