Thank you. This is what I’m trying to get at. How long does it take to do justice to each city? Are these cities where you could spend an unlimited amount of time (like a Paris or Rome), or are these cities in which 2 days will do it?
And what do you mean by “Norway in a nutshell”?
What is in Bergen, specifically?
Appreciate your apparent European expertise 
If you’re in Sweden in mid-July you could take the Helsingor/Helsinborg ferry crossing then head up the coast, stopping in Bastad for a day to watch a bit of the Swedish Open tennis tournament. Bastad’s a beautiful small town with sandy beaches and a thriving summer culture and during the Swedish Open you’ll see the glitterati of Sweden. It’s directly on the road to Gothenburg or a half hour detour off the route to Stockholm. Even if you are there after the tournament’s finished, if you do go up the coast it makes a nice place to stop for lunch or a swim.
I love Gotland, but I think it would be out of your way on this trip. Stockholm is a beautiful city with lots to see but I’m not sure I’d want to do all the driving necessary to see both Oslo and Stockholm. I’d choose one or the other. I love Stockholm and I think it fits your description of what you want, but I haven’t been to Oslo so I can’t compare the two.
My general advice is not to try to cram too much into your trip or you won’t have time to wander the streets, people watch and absorb the culture. Instead of thinking of this as your one and only trip to Scandinavia think of it as your first trip. That way all the things you can’t get to on this trip won’t drive you crazy. Just think, “We’ll do that next time.”
Personally, you could spend a bit more time in Copenhagen, but none of those cities are like Paris, Rome, Berlin, etc, where you could spend weeks and not cover everything you wanted. Ideally, I would suggest spending 3 days min, but it’s not possible for you, and I couldn’t recommend 86’ing one of the countries. 
Norway in a Nutshell is a train from Oslo to Bergen that has a spur where you take a mini fjord cruise. One could do it as a day trip starting from and returning to Oslo, or, as I suggested, starting in Oslo and ending in Bergen. Either way is fine. Bergen is a cute Norwegian city, but if you just wanted to return to Oslo after the fjords, that’s fine as well. It’s just that makes for a very long day.
Two to three days in each city are probably fine. But I think Norway in particular is a good one to really get to other places , Bergen, the fjords, etc.
Not to answer any of your other questions, but… my husband, who is well traveled, but doesn’t want to go anywhere a second time, keeps saying Stockholm! We gotta go back to Stockholm! I haven’t been, but for him to love it that much is significant.
It’s very easy to get around in Stockholm, really in any of the three cities. Lots of places to walk to. We even took a city bus with locals in Copenhagen with no problem. The bus driver even made a point of telling us where exactly we should get off the bus as he knew we were trying to get to Rosenborg Palace. (very interesting, by the way).
@Sue22 I agree about both Goteborg and Stockholm not being feasible on the same trip in addition to Copenhagen and Oslo. I only brought up Goteborg because it sounded like Sweden was possibly going to be left out completely due to time constraints and was suggesting a way to at least see a bit of Sweden.
Agreed. Public transportation in all three cities is amazing. In addition, each city has an efficient train system from the city center to the airport.
I will also tell you that the Scandinavian countries are one place where you don’t need a phrasebook; everybody under 90 speaks English, and while they are all very helpful, they’d much prefer it if you did not try to butcher their language. No bonus points for even trying.
You could see quite a few of the main sites in Copenhagen in 2-3 days.
“you don’t need a phrasebook.” Agreed. We only came across one person the whole time that spoke NO English at all. An elderly man in Copenhagen out with his dachshund. Since we had a dachshund at home, my sister and I started to talk to him and it was obvious he had no clue what we were saying.
Oddly, the bus drivers in Helsinki often do not speak much English (I know the OP isn’t going there, but just saying – spent over a week there once and found it to be common – odd because almost everyone else we met spoke it).
“I only brought up Goteborg because it sounded like Sweden was possibly going to be left out completely due to time constraints and was suggesting a way to at least see a bit of Sweden.”
No, I haven’t made a decision on what to leave off. That’s why I’m asking all you nice folk!!
Copenhagen is the only one that is definite then? Since you said initially you wanted to go to Denmark (Copenhagen) and either Norway or Sweden. Personally, if I had to leave one out, my thoughts would be 1. Norway 2. Sweden 3. Denmark But, everyone has their own preferences. They are all great so you can’t go wrong. But, as others have said, if you can find a way to do all three, that would be great.
Yes, Copenhagen is the definite here. I thought I put that in my OP but apologies if I wasn’t clear.
Pardon me for not having a good map at my finger tips. I LOVE Copenhagen. It is the best blend of old world and ultra modern Danish design. The people are wonderful and the food is amazing. A few favorites things would be head up to the Louisan Modern Art Museum and then to Helsingor and see the Hamlet castle. A drink or two on the Nyhaven canal is a must, the tour boat around the harbor is interesting, Tivoil if it suits you, wander over and say hi to the Little Mermaid and watch the changing of the guard at the palace. Just west of Copenhagen is Roskilde with a beautiful cathedral and the fabulous Viking museum right on the fjord. Drive south and see the Mons Klint (white cliffs)
The only place I have been in Norway is Tronheim which offers an amazing vista on the fjords. I have heard from many, many people that Bergen is more enjoyable than Oslo. I have seen cruise ships heading out from Copenhagen to the Norwegian fjords. That would be stunning. I think I would at least take a day trip from Bergen up the shore.
I have been to Stockholm several times. I find the city fascinating and very hip and the restaurants are quite cosmopolitan. The Vasa Museum is very interesting and really not out of the way. I think everyone our age should book a tour of the Abba Museum. Where else can you dress up in one of their costumes and lip sink on stage! The Nobel museum in Gamla Stan is very interesting. A boat ride to Drottingholm (the queens summer palace) is a great way to spend a couple hours. Uppsala is just north of Sweden. It is a college town, has a stunning cathedral and also the the home of the Linneaus museum (the gentleman who gave Latin names to all of our plants)\
We have always toyed with taking the multi day boat ride on the Gota Canal between Stockholm and Goteborg.
Denmark and Sweden are both super easy to drive in. We have also taken the train from Malmo to Stockholm. I think is is about a 5 hour ride.
If I had a 10 trip planned to the Nordic countries, I might consider flying Icelandic air and spending 3 days in Iceland before continuing on to one one of the others.
Yeah, @Kajon , One of my first thoughts was Icelandair and stopping in Iceland ( my son did this last year) but @Pizzagirl just doesn’t have the time. It sounds like a great trip. You will love Copenhagen. Just need to decide what other places to hit.
We visited Stockholm and Copenhagen. Visit the vasa museum in Stockholm. We took the sleeper car train to Copenhagen and the ferry over for a day to Malmö. Loved biking in Copenhagen , people watching with a coffee or glass of wine in the outdoor cafes and going to the Royal Copenhagen store.
PG said summer or early fall. I wouldn’t go to Sweden in July. That is their national summer holiday and many things close. Even hospitals close. Trains are on a lesser schedule, etc. Of course, the weather is really nice then!
You could go in late June and have the longest days and the big holiday of Midsomer. August still has very long days and good weather.
My s and dil are heading to Iceland next month and just chatted with a friend who returned from there. All I recall off the top of my head was the recommendation to eat at Dill in Reykjavik, and to take the golden circle tour to the secret spa rather than the blue lagoon spa which they said was too expensive.
“Oddly, the bus drivers in Helsinki often do not speak much English…”
In my experience English is not quite as universal in Finland as it is in Sweden and Denmark. Young and/or well-educated Finns usually speak excellent English, especially in Helsinki. But once you get off the main tourist beat English gets increasingly rare. In a small to medium-sized town away from Helsinki, such as say Kotka, almost no one speaks English. In places such as that Swedish rather than English is the main second language.
I just read the first few GPS and see that the vasa museum has been mentioned several times. It’s really cool. We also heard a concert in Tivoli Gardens in Copenhagen. IIRC we stayed very close to the gardens so could walk, but my memory is rusty… Will double check with DH. I could be mistaken. But we did not have a car in either city.