Let's see what can be done. Having read your comments carefully some specificity and altering of the recommendations is in order. Here is a proposed itinerary. You will have to choose the exact number of nights depending on how you feel after having had a quick look at the destinations. Look them up and gaze at photos. 60 seconds per destination will usually be sufficient.
Arrive in Copenhagen (4 nights). Visit some nearby cozy coastal villages, such as Dragör and Helsingör to the south and north, or have a look at the Köge village with its sandy bay. While in town consider Rosenborg Slot (castle) or Louisiana (museum of modern art) which must at times be the best museum in all of Scandinavia. The coastal setting certainly enhances the experience. There is also Ny Carlsberg Glyptotek for older art. Take an international commuter train (frequent departures) to Malmö/Lund (2-3 nights). Stay in Malmö and day trip to Lund at a moments notice on one of those frequent commuter trains, to experience the interiors of the old cathedral (photography allowed) and its astronomical clock which plays twice daily.
THE NORTH
Go north on a long distance train towards the capital, stopping along the way to see a quaint old wooden village; Eksjö, or a small medieval town; Söderköping, before taking a ferry from Oskarshamn to Gotland and Visby (2-3 nights). Take a ferry to Nynäshamn when returning to the mainland and then a commuter train (frequent departures) into the capital (4-5 nights). While there bring the camera to City Hall, the Royal Castle apartments, and the inner city island of Djurgården, with its open air museum, fields, heritage houses and several art galleries. Day trip to Uppsala for a look at the largest cathedral in Scandinavia (medieval), the old castle up on the hill, the extensive 18th century gardens at the foot of the hill and the bronze age viking burial mounds just outside town. Day trip to Gripsholms Slott (castle) and Mariefred, the quaint adjacent village. Then go southwest by train to friendly Gothenburg (1-2 nights) and depart from Landvetter Airport (GOT).
OR
THE TRUE NORTH
Go north on a long distance sleeper train to the foremost of the provincial capitals of the north; welcoming Umeå (2-3 nights), a relaxed frontier university town, close to the coast, set beside a wide stream and surrounded by fir forests reminiscent of the Canadian wilderness.
THE NÄRKE PROVINCE
Go west to inland provincial capital Örebro (1-2 nights), with some of the most pleasant and amiable people in the country, the truly imposing Örebro Slott (castle) and a favourite city park with an open air museum and the delicious Jeremiabageriet (heritage bakery). Nearby is the quaint old wooden village Nora.
THE DALARNA PROVINCE
Go northwest to old-fashioned lakeside Rättvik and Mora (2-3 nights), end of the line inland villages in Dalarna, and appreciate the world-class art of Anders Zorn at Zorngården (museum), master painter of Mora, who came from very humble beginnings and went on to win the gold medal at the 1889 world’s fair in Paris and was honoured by France with the Légion d’honneur. One of the 3 most important artists ever to come out of Scandinavia. Have a look at his portrait Queen Sophia (1909) .
