There are so many places out of the center of Copenhagen that it would be worth renting a car for a few day trips.
Two day trips I'd recommend,
Dragor and Kronborg are worth a visit.
Along the way there are wonderful landscapes and other lovely towns to explore. Danish public transportation is impressive and most likely will take you to any place mentioned here. However, getting around in a car is really more convenient for those spur of the moment stops.
Driving to Copenhagen from Copenhagen around Copenhagen has been hell, I live here but only had car for the last week or so, but even though, I have been sitting in a car for long time here, but just the other day I went from downtown Copenhagen to my communa Gentofte and I drove around it, by it over it and god knows what else for over an hour before I finally found my way home.
So after 2 days my boss gave me a GPS and fuff, what a relief, so I recommend to all of you if you are going to drive around use GPS :-).
Another thing you should also pay a good attention to are that people on bicycles do they always have the right, and you get a big fine, and bigger fine for driving them down and a bad conscience.
Traffic across Copenhagen has seemed to us as impossible free. We reached the city centre for 10 minutes and left the city during 15 minutes. In Moscow it is absolutely impossible! Stockholm left the same impression. How easy is it to drive in European capitals!
We enjoyed by freedom of movement in all northern European capitals - Helsinki, Stockholm and Copenhagen. About traffic in small cities - Turku, Uppsala and Helsingor nothing to speak about - it seemed to us they were almost empty from cars!
This is just one of the many wierd little cars that they have here in Copenhagen. They say this is big and roomy but if ya come from the states it's a small egg. Almost all cars here have to be small though because of the gas prices.