travel budget for family of 4 over 3 weeks in scandinavia

That's an expensive area, though the most expensive Scandinavian country by a considerable margin is Norway.

The five major expense categories for any American taking a trip to Europe are transatlantic airfare, lodging, food, intra-European transportation and sightseeing.

It's hard to comment on transatlantic airfare costs, because the ground has been shifting since early 2020 and costs will vary a lot depending on your home airport(s). The best thing you can do at this point is price out multi-city flights (into one city, out of another) for 2024 dates that approximate when you might want to travel either next year or in 2025. Keep notes; knowing the range of fares will help you spot a bargain if one comes along after you are ready to buy tickets. I use Google Flights, but if you already have a flight website you like to use (like Kayak), it will probably work just fine. Ignore wacko offerings from sources other than the airlines themselves. (No CheapOair, etc.)

For lodgings you can explore Airbnb and the non-hotel offerings on booking.com. If you need to use hotels in some locations, you may save 5% or more by joining a hotel chain's affinity club. I usually stay in cheap, non-chain places, but it wasn't so easy to find places like that in Norway, Sweden and Finland. (I haven't been to Denmark for 50 years.) Peak-season hotels can be quite a bit more expensive than off-season deals, but some business cities (including Oslo) seem less busy in the summer, so comparative deals may be found. Be sure to look at summer prices and be as accurate about dates as you can. I found significant rate increases between 2022 and 2023 when I revisited some of the same cities, so be aware that rates mentioned in guidebooks or by folks who traveled prior to 2023 may be quite out of date.

Food costs are tough to predict because people's preferences and standards vary a lot. Sit-down restaurants in Norway, in particular, are punishingly expensive. The labor rate is extremely high, and that is reflected in restaurant prices. Less-expensive options in Scandinavia include buying things like cheese, yogurt and fruit at grocery stores; pizzerias; ethnic restaurants and order-at-the-counter/take-out places. Google for information on local tipping standards so you don't fritter money away by tipping like an American (which is unnecessary). Alcohol in restaurants can be very expensive, so check on that ahead of time. A lot of restaurants post priced menus on their websites. You can go to Google Maps and look for restaurants a few blocks from sights you know you want to see, click on their icons and follow the Google link to a website. That will give you some idea of what you're looking at for prices in a part of the city where you will probably be at some point.

Inter-city ground transportation will generally be more expensive than farther south in Europe. Both train and ferry tickets (I'm not sure about buses) may increase in cost as you get closer to the travel date. In other words, there's a potential to save some money if you buy the tickets early--but check the change/cancellation rules. There will be affordable public transportation to and from airports; be very careful about taxis, which are likely to be highly costly. Taxis in Norway are mortgage-the-house expensive.

You'll probably need to use a good bit of transit in Stockholm. It's a beautiful city that really repays sightseeing on foot, but it's spread across a bunch of islands (albeit with bridges), and you'll have two children with you. Single transit tickets are quite expensive; passes are cheaper.

Entry fees won't be super-cheap, but Scandinavia doesn't get the flood of tourists heading to places like Italy, France and England, so sightseeing costs may not be shockingly high. The children will be either free or much less expensive. The best tip I can give you is to see what you want, not what's on the Top 10 lists.

Consider renting a car for a part of your trip. You have the freedom to go to places where public transportation is very limited and slow. If you are a family of 4, it is also more convenient. Rent a station wagon to fit your luggage.
Have you considered camper van for a part of your trip?
It's a car and a hotel in the same price!
Kids find that fascinating (maybe not all, but most of them)

Keep in mind that although it will be warmer in the summer, it may not exactly be warm. I was on the Rick Steves Best of Scandinavia in 14 Days tour the 2nd two weeks of June, 2018. I needed to layer everyday, often including both a warm jacket and a rain jacket.

Go to timeanddate.com (https://www.timeanddate.com/) to see not only averages, but detailed day-by-day historical weather data for the dates you plan to be in the locations you plan to visit. There's so much there, it's more than a bit confusing. Here's how I got to the past weather data for Copenhagen linked below. This may be more than you need or want to know for your trip, but here goes.

Click on Weather. Scroll down to Copenhagen and click on it. Click on Past above the Now box. The Select month box to the right will default to the Past 2 Weeks. Click on the down arrow next to that box. Scroll down through the options there and choose a month and year. I chose July 2019 to get this result. There's a detailed graph with data that will change to any date you might choose below it. You can also scroll the graph right or left to the dates of interest to you. Below the graph is summary data for the month as well as even more detailed time of day numbers than are shown in the graph.

Beyond the weather, as part of your research you might want to take a look as the Explore Europe section of this RS website, especially the parts on Denmark and Sweden linked below. They include good info on both Copenhagen and Stockholm.

Explore Europe.

Denmark: https://www.ricksteves.com/europe/denmark.

Sweden: https://www.ricksteves.com/europe/sweden.

This "when to go to Scandinavia" article might also be helpful: https://www.ricksteves.com/europe/best-time-to-go-to-scandinavia.

So should this RS recommended itinerary depending on how much time you have. It includes driving as well as public transportation options. Note that the itinerary has more locations for Denmark than for Sweden.
https://www.ricksteves.com/europe/scandinavia-itinerary.

As you read and research more you'll definitely be able to see what sights and experiences would be best for the kids when you're there. Something else you might do as you plan is to take a look at the daily itinerary for the tour I mentioned above. Its details might give you some ideas. Here's the link: https://www.ricksteves.com/tours/scandinavia/best-scandinavia-tour.

It looks like this may be your first trip to Europe. If so, it would probably be a good idea to devour the RS Travel Tips linked below. For me they are a great resource for learning about things I didn’t know I needed to know. https://www.ricksteves.com/travel-tips.

Beyond the high prices, one annoying thing about the Scandinavian countries is that they each have their own currency. One great thing about them is that you can use a credit card for just about everything.

Using Booking.com to search for apartments for your family will give you some idea of what is often the most expensive line item in Europe. When you do the first search, you'll fill in the number of adults and number of children. Then it will ask for the ages of the kids. Your kids will be young enough to not be considered adults. If you filter for 2 bedrooms, pay close attention to the results. Beds may be a large bed created by 2 beds put together which can be separated into twins or a double that is not as big as a queen. The 2nd "bedroom" may be a sleeper sofa in the living room.

If you're coming to Sweden/Denmark, the warmest months are July-September. However, evenings do get chilly. Have sweaters for you and the kids.

I figure, aside from airfare, $250 a day at the very minimum is what you will need.

When renting a car, you will need to rent car seats for both children. Even if you take a coach bus from one city to the next, you may be required to have some sort of child seat. I saw a woman with a toddler who was almost denied boarding on a bus because she didn't have a seat for the child. Fortunately, a seat was found on the bus that she could borrow. Just something to think about with small children.

Airbnb is not that common for apartments in Sweden because of legal issues. No idea about Copenhagen. Renting a cottage is easier but they can also be in high demand. Book early. Watch out for scammers. You might rent a house that doesn't exit.

Bicycling in Copenhagen is fairly easy but a bit crazy because so many people do it. You can get by using trains and buses almost anywhere. Just be careful about checking to see when the last bus leaves if you go to a small town. It could be very early, like 5-6 pm. When you've got the long days of midnight sun, then your sense of what time it is gets skewed.

In Sweden and you can use the SJ train app to book tickets. You can book to your final destination even if it's not serviced by trains. SJ sells you the local bus tickets in the price as well. For Denmark you can download the DSB app. However, you can book Copenhagen to Stockholm on SJ.

Be sure to book early. Flights, accommodations, train tickets. They are subject to price hikes depending on demand.

This is a beautiful area in the summer. I am sure you will enjoy your vacation here.