There are many cute cities and towns in Denmark. One of them only 15 minutes from Copenhagen is Lyngby. It is a very picturesque place often full of pedestrians. There is a Main Street full of shops. There is also a bigger shopping centre and Magasin du Nord. You can also visit a great furniture shop Ilva- a good complement to IKEA!
If you want to explore areas outside of Copenhagen, I truly recommend a visit to Lyngby where you can stroll around, window shop and shop and sit down and have something nice to eat when you start to get that little bit peckish!