Gothenburg

It's helpful to when this trip is.

1. What do your children like to do? A bit more information would be great. But in general, Liseberg is usually popular among children, www.liseberg.se . Gothenburg has some great museums as well, have a closer look at Universeum www.universeum.se . A visit to the archipelago maybe? Or just a boat trip to see the city from the water. In the summer there are tour boats that run on the old moat, and year round there are regular ferries on the river. Or Maritiman if they want to see old ships, http://www.maritiman.se And later this year the new Volvo museum opens, www.worldofvolvo.com

2. Does it have to be Copenhagen? Stockholm is also pretty close. But there is a lot to see in the area as well. Again, what are you looking for and what time of the year is the trip?

Month and Year? Home state?

You may want to look into the famous "amusement park" in Gothenburg, Liseberg. It's not at all like Disneyland or even Tivoli Gardens. It's more like (maybe) a classier Knott's Berry Farm, but with culture and few rides. More of a recreational day-out than ... Pirates of the Caribbean. It was a must-see for us, and we're not roller-coaster people or even theme park nuts. (We NEVER go to Six Flags in our home state, NJ.)

Edit: The Gothenburg Symphony Orchestra is an internationally noted ensemble.

We live in NC. My kids are pretty versatile. They like riding go-karts, going to arcades, hiking, but they also loved the Prado. I would say, the biggest draw for them us anything they can see (or anything I can sell) as an adventure.

We are thinking June or July but haven't nailed down dates yet, partly because we're still figuring out where to fly in and out.

Copenhagen isn't a definite, but hubby thought the kids would like the train over the bridge.

He will be in Gothenburg several times per year, and we're hoping to join him at least every year or two, so we also don't have to do everything all at once. The more I read the more I think we may want to just explore in and around Gothenburg, maybe take a day trip to the archipelago and a day trip to Varberg. The hardest part will be figuring out what things he doesn't mind missing out on during his work days.

If you want something you can sell as an adventure, a trip to the archipelago seems like a good idea. Especially the southern archipelago is very easy with ferry routes to most inhabited islands. A day trip to Varberg and a visit to the fortress is also a great idea. For a summer trip, there is a preserved railway not far from Gothenburg if you want to take a trip with a steam train. (https://www.youtube.com/watch?v=preDhYFThlc) And the museum of natural history might also be fun for them.

Regarding where to fly to, Gothenburg has an airport that is connected to most major hubs in Europe so you can flights there if you want. But Copenhagen and Stockholm are also options.

If you cannot get a direct flight to Gothenburg (SAS had them in the summer last year on certain days of the week), you will have a connecting flight from Copenhagen, Stockholm or Oslo. So you can add a couple of days at the beginning or end of your trip and explore those cities. Copenhagen: Tivoli and the Louisiana Museum. Oslo: Vigeland Park. Stockholm: I like Skansen (open air folk museum with elk!) Does your husband work for Volvo?

There are lots of things to do in Gothenburg. Even riding the trams will be an adventure for the kids. Paddan is a flat boat that gives tours of the city. There is a vintage Liseberg tram that runs during the summer months. Kids and adults love that.
Even just walking through the city has its charm. You can visit Trädgårdsföreningen near the city center for a lovely walk, but there are also the Botanical gardens (https://www.botaniska.se/) and Slottskogen, which are both very large parks. In fact, you can go hiking in the large forested area south of the Botanical Gardens (3.5 km X 2.5 km), which is surrounded by the city and the suburbs. Want something even bigger? Try Delsjöområdets nature reserve, which is very close to the city center. Just over a kilometer from the Liseberg amusement park. And, for the price of a tram ticket, you can ride on of the harbor ferries on Göta Älv from Stenpiren. There are also trips to the islands on the archipelago. Just make sure you check the time table carefully so that you don't miss the last one back. (I have run into tourists who have misunderstood how to read the local bus schedules.)
Of course, there are a variety of museums, from the Volvo museum to the Sjöfartamuseum (Maritime Museum and Aquarium) in the harbor to the Natural History Museum in Slottskogen Park. The Gothenburg City Museum along one of the canals gives the history of the city. It is in the building where the Swedish East India company was housed. The East India Company was very important for the prosperity of Gothenburg.

The natural history museum has a small blue whale that might interest the kids: https://www.gnm.se/en/exhibitions/permanent-exhibitions/the-whale-hall/