Sounds lovely.
In Denmark I would say Aarhus or Copenhagen are the obvious cities. Copenhagen due to a lot of historical places, beautiful palaces, the little mermaid, two of the world's oldest ammusement parks (one being the oldest) and several fine museums.
Aarhus isn't a city that you can spend a lot of time in, but it's defenitely worth visiting for a couple of days. We have a great arts museum with lovely views from the rooftop rainbow panorama, the lovely old town, where you can see houses from different eras and Moesgaard Museum that is a highly interesting cultural history- and archeology museum. Also there are lots of historical areas and buildings as well as some lovely spots of nature including the bothanical garden and the park that is a part of what's often called one of the world's 10 most beautiful universities. Also Aarhus is most likely where you could find me
Other places could be Roskilde or Ribe. Both are smaller, but very historical.
In Sweden I would always say Stockholm. Though it's an expensive city, it's really beautiful and full of interesting places to go.
If you want some good ideas for less expensive cities (both Denmark and Sweden are decently expensive countries) I would say that Cracow and Warsaw (), Prague () and Budapest () are all very interesting. Berlin is a bit more expensive, but still very payable and also highly recommendable.
If you want to go to southern Europe, my main recommendation would be Florence.