I taught in Denmark for two years 15 years ago and have been back to visit many times.
My vote is for Helsingborg via ferry! You can visit Hamlet's Castle and then catch a ferry (the cheapest prices on candy!).

My daughter (12 years old at the time) took a bus then train from outside Copenhagen with her U.S. friend up to Helsingborg, then to Sweden on the ferry. They spent the day wandering around, having ice cream, etc. Then back on the ferry. Riding a ferry is so Danish!
Have a great trip!