Wonderful experience to one of many baroque castles in Denmark which provides excellent history of Kings and Queens in the Renaissance era at the beginning of 1600s by King Christian 4 which at one time dominated Europe. Castle is surrounded by lush forest and gardens that features in laid waterfalls and boxwood patterns that identify the current Queen's signature crest.
Hillerød is approximately 35 minutes North of Copenhagen and can be reached by S-train Hillerød. This quaint town has very good restaurants and shopping accommodations to meet your desires once your visit has been completed.
If you will be in Copenhagen for several days, would strongly suggest purchasing "Copenhagen Card" which provides access to Museums, guided tours and rail/bus transportation services within a wide area of the city. Value of card will vary from $100--$300 person depending on length of time spent in Copenhagen. It will be money will spent based on my recent experience.