It is about a 30 minute ride. There you will be able to visit he Viking Ship Museum or visit the more than 800 year old Cathedral, which is the burial place of most of the Royalty of Denmark. You may also chose to go shopping or sightseeing in the Danish Capital. The Royal Copenhagen shop, Illums department store and much much more. A dinner is recommended at one of the many waterfront seafood restaurant, or at a local place such as Parnas is also a good recommendation.
Prices there are reasonable, and the local dishes are great.
After breakfast you may wish to do some more shopping in Copenhagen, possibly take in a nice lunch at one of the many small and cozy restaurants near the walking street. You may also wish to take a day tour by train to Odense, the home city of Hans Christian Andersen, and visit his house and museum. You could in fact go all the way to Legoland, which is only an hour beyond Odense by train. This can be done as a one day excursion, however it can also be arranged as an overnight trip. After breakfast the day is on your own.