What's good (at least the parts I liked the best):
Helsinki, the island fortress Suomenlinna. You can take a little ferry from right at the city center. I also liked the city center. The rock church was skippable for me. It earned a meh, from me, and not worth the effort to see it.
Tallinn, the old center.
Oslo, the Viking ship museum (by ferry from the center), I also really liked Frogner Park (sculpture garden). Walking the center including the city hall and opera house were nice (both worth a look, not major sights for me). I didn't tour inside Akershus because the people with me weren't interested.
Stockholm, I liked the Vasa best and wished I'd spent more time on Djurgarden. Was not impressed, however, with Skansen. I guess I was expecting it to be like Williamsburg, VA where you can ask the staff about things - and they can tell you. In Skansen, the local guests knew more about the buildings than the staff did (e.g. how were farm houses oriented and why?). I liked walking around Gamla Stan, but it's only worth an hour or so.
Copenhagen. One day trip to Roskilde to see the cathedral and Viking Ship Museum (completely different type of place than in Oslo) was worth it. Look for the bones of Herald Blue Tooth built into the wall of the cathedral. Admission and transportation was free with the Copenhagen card. Another day trip was up to see Helsingor and the brief but good castle tour (take a flashlight for the casemates or buy a cheap souvenir from a vending machine there), then a "private rail line" further up to a beach at Hornbaek to get sun and wade in the Baltic. I didn't see Frederiksborg, in Hillerod, because they were working on the rail line and it would have been really inconvenient. If I go back, it will top my list of to-dos.
In town, I liked the National museum (tons of viking and medieval things), the Glyptotech sculpture gallery, Rosenborg slot is really nice and has the Royal Jewels in the basement. I also liked the Nyhavn area for staying in, the canal cruise from here is worth taking. Nyhaven is cleaner and nicer than around city hall.
City Hall is worth a look. I ran out of time before visiting Christiania, so can't give you an opinion. Amalienborg has the changing of the guard and some ruins underneath, a meh from me there. Tivoli earned a meh from me also. I kept thinking Danish kids really aren't overstimulated. The average American kid wouldn't think much of Tivoli. I did like a restaurant there (name translates to the Ditch) - we ordered Smorebrot for four.