What to see? What to do?That really depends on your interests. As I'm a culturally challenged wildlife-type of Swede, I might not be the best person to ask though. However, my favorite tourist places in Stockholm are the 'Old Town' with the Royal Palace (with embedded museums) and the Vasa Museum. Check this page.
Agreed. I would add Skansen, which is a combination open-air museum (restored buildings and crafts)/ theme park/ zoo/ outdoor concert place. Another must-see is the Stadshuset, the Stockholm City Hall, which is very interesting in its architecture and interior design and decor.
My wife and daughter and I will be in Sweden again for two weeks, August 23-September 5. We're spending the first week in Stockholm, and then the second week on the West Coast, to visit family and friends and for me to go to a theological conference in Gothenburg.
BTW, you may find it helpful to know the Swedish name for the "Old Town," which is Gamla Stan.