I've never been there, but there are a couple of places in the world that if you need to ask for suggestions on what to see/do, you just shouldn't go in the first place.
I've been to Oslo, Helsinki, and Stockholm many times. Unfortunately, I haven't been able to see much of the region outside of those cities. While I like all three cities, Helsinki is probably my favorite. It has a very laid back culture. Stockholm is more cosmopolitan, but there's a very old section of the city that is very interesting to visit. Oslo is probably my least favorite, but I've been there less than the other two and mostly in the winter when the sun barely gets above the horizon. Outdoor activities are popular in the whole region, and many people own vacation cabins around lakes, etc. There's not nearly as much in the way of cultural sites (art museums, etc.) as other parts of Europe.
I spent a month in and around Stockholm several years ago. Stockholm in the summer is hard to beat because essentially it's a city of islands. The Old Town (Gamla Stan) was typical Old Europe (narrow streets full of bars and restaurants), very nice.
Weather was fantastic (70's in July) and the women, of course, were gorgeous. It was expensive back then, so I can't even imagine how expensive it would be now for Americans.
We also took the train down to Lund in southern Sweden which is a cool little university town. Lund is just across the bay? to Copenhagen which was fantastic. There's a little hippy commune there called Christiana which sold whatever you wanted right out in the open--
but I think I've read that in recent years that the cops have tried to put an end to that.
Anyway, I loved it over there. The people were nice, if not a little reserved, but nothing several shots of Absolut couldn't fix.
I would love to go again but definitely couldn't afford it these days.
I did a Norwegian fjords cruise at the end of last summer. That may be the most beautiful place I've ever seen. For those who have been thinking about taking an Alaskan cruise, I would suggest ditching those plans and making the once-in-a-lifetime trip to Norway instead.
I've only been to Aarhus, Hillerod, & Copenhagen in Denmark; Oslo, Bergen, the Flaam river valley, and a fjord or two north of Bergen in Norway; Stockholm in Sweden; and Turku and Helsinki in Finland.
In Norway, skip Oslo, go to Bergen. Beautiful city, very walkable, lot so young people and energy. Can't beat it in the summertime.. it stays light until about midnight.
Bergen is also a great launching pad to explore the Fjords. If you did clear cool water, dramatic mountains, and heavily wood forests, that is the place to be.
When I was young I lived in Oslo for about half a year. It's been so long any references to the city would be dated. However, I want to emphasize the natura beauty of Norway. We spent a lot of time x-country skiing and traveling through the country; much prettier than Sweden or Denmark. In fact, moreso than anywhere else we went in Europe save perhaps the Alps.
We also took a ferry once up the coast to Bodø and drove back, that combines the best two ways to see the fjords.
Honestly, I'm planning on going back and backpacking in their national parks.