Frederiksberg is an area of Copenhagen where I have been spending more and more time during my last visits. It's a more residential area of Copenhagen and it's full of great cafes and shops.
The main reason why I have been coming here has been the beautiful park Frederiksberg Have and the charming Cafe Funder which are located here, but there is a lot more to come for.
During summer time you can spend your Saturday morning at the local flea market and there is an indoor swimming pool located in Frederiksberg.
Frederiksberg is right next to another very nice - although a bit more urban - area of Copenhagen, Vesterbro, which you may just as well visit once you are here.