It is a popular belief that Scandinavian countries are expensive, and whilst that is mostly true, visiting Copenhagen does not have to put a huge dent in your wallet. Admission to the many museums is very pricey, but you can purchase a Copenhagen card gives you free admission to 60-plus attractions, plus discounts on many more. It also includes public transportation. It is available for 24 hours or 72 hours.
There are many worthwhile and historic things to do whilst in Copenhagen that include a visit to the Fredericksberg Castle and gardens and the Christiansberg Palace. Also, included in the price of the pass is unlimited visits to Tivoli gardens, which is a fun and interesting amusement park.
If you wish to visit fewer attractions, I would definitely recommend the Fredericksberg Castle and gardens and the Tivoli Gardens, both of which are beautiful. The Hans Christian Andersen museum is also interesting both for adults and children. I think Copenhagen is definitely one of the hidden gems of Europe and is a wonderful place to see, visit, and experience.