Europe in April with Family

You could go into the RS tour section on this site and see which countries have the least amount of rainfall as well as the average temps for the month. We did Greece in April of 2018 and the temps were cool and only had a couple of days of hard rain. (Of course that was a few years ago.) You might keep in mind that the further north you are the lower the temps and the greater the chance of rain in April. For example, they show Brussels for April with an average high of 58, low of 48 and 12 days without rain.

Central Europe is relatively dry year round. I have maybe 6 years total time here I have yet to be caught in a downpour like we have in Texas. But you will have to look up the numbers for whereever you are considering. For that I suggest you also look up your hometown so you have a benchmark. After all, what does 1.2cm total with 12 days of rain possible even mean in real terms.

Three cities? Then you named two thirds of a good tour: Prague, Vienna and Budapest. They line up neatly, are each different but share a common thread. Its a very popular tour. Or replace Vienna wiht Salzburg to have little smaller destination on the trip. And there are a number of smaller but interesting one night stops along the way if that interests you. Normally I say London and Paris and Rome if someone has kids at an age where they can learn, but yours are a bit young so this will be less learning and more fun.

If you want more outdoorsy sorts of things, then Croatia and Montengro and maybe Bosnia. Croatia isnt at the top of my list (too many tourists and too expensive) but many love it. Montenegro and Bosnia on the otherhand are pretty amazing if you love nature, unique towns and cities and just overall uniqueness. Look at the videos on this thread: https://community.ricksteves.com/travel-forum/general-europe/back-doors-to-the-back-doors-of-europe

March temps might be a tad cool in Prague, but more moderate in Budapest and still warmer the further south you go.

Or, if you dont mind a flight and want to mix it up, travel light so WizzAir is cheap and do Budapest and Montenegro and Bosnia. Budapest to Podgorica, Montenegro is about 75minutes and as low as $45.

Your big cost is getting across the atlantic. After that follow your heart.

How many nights will you sleep in Europe? Arrival day is about settling in and nothing more so don’t expect to do any sightseeing that day.
Since you’re going in early spring save the Nordic countries for a summer trip. Instead, narrow it down to the Netherlands, Prague and Budapest. Keep in mind you lose a day when traveling from one place to another.
Amsterdam might be easiest to fly in and out of. You could spend all your time there and take day trips which would be easiest since traveling with eight-and-twelve-year-olds. If not, consider flying multi-city in and out of Prague and Budapest. Most likely there will be more connections than flying in and out of Amsterdam since these cities have fewer flights in and out of the US.

Amsterdam is great for this age. The Anne Frank House has a great presentation that even a typically developing 8yoa child can grasp. It will definitely help to read the book (or otherwise learn about Anne and the holocaust) before the trip. The VanGogh museum is also excellent. Both require advance booking, as I recall. My 12yoa son also enjoyed Those Dam Boat Guys' evening canal tour (bundle up, though). The fries!
From Amsterdam, we went to Bruges and did a day trip by train to see the highly recommended Atlantic Wall Museum (after which we walked on the cool, windy beach at DeHaan--again, the fries!).
Copenhagen was another favorite for my son. Tivoli is lovely and fun. We had a friend's car and "hunted" for Thomas Dambo's troll sculptures.
My standard advice is to determine the "best" transatlantic flight for you (best being very subjective, could be cost, schedule, leg room, air miles, whatever), and work from there, using seat61.com to help with train connections and skyscanner.com for flights.

With 10 days, you could do Prague, Vienna and Budapest. All great cities and you would have three days in each.

Amsterdam and Netherlands, perhaps adding Brugges and Ghent, would be great. Amsterdam has the Anne Frank House, the Rijksmuseum and Van Gogh Museum. Make reservations early for the Frank house. Also, consider a day trip to the countryside, like a Windmills tour.

I don't recommend Scandinavia until Summer.

One thing, I lived overseas and traveled with children one below 10 and one a pre-teen. Especially for the 8 year old, museums many not go well. I remember my 7 year old son after we were in the Lourve for 45 minutes, asked "are we ready to go."
I suggest having you kids read up in advance on the history of the places you visit.

With 10 days, you could do Prague, Vienna and Budapest. All great
cities and you would have three days in each.

Yes, but its pretty fast paced.

Day 1 Arrive jet lagged
Day 2 Prague
Day 3 Prague
Day 4 Long morning train to Vienna
Day 5 Vienna
Day 6 Vienna
Day 7 Short morning train to Budapest
Day 8 Budapest
Day 9 Budapest
Day 10 Depart

The OP didnt mention Vienna, so there are other options.

Day 1 Arrive jet lagged
Day 2 Prague
Day 3 Prague
Day 4 Prague (day trip day. plenty of ideas from our Prague experts)

To be honest. Prague is great. But for my taste its either a two full day stop where you do what most tourists come to see and do, or you make it a week and do a different sort of trip.

Day 5 Flight to Budapest. This can be fairly cheap depending on how you pack. Shaves about 3 hours off the trip vs the train. Or take the train.
Day 6 Budapest
Day 7 Budapest
Day 8 Budapest (day trip or overnight trip. If this works for you we can put together a pretty good list of interesting options from city to town ... maybe Lillifeurd for the forest railway and the caves and nature? - I love that idea as it is change from city things - a couple hour train ride each way)
Day 9 Budapest

Why the extra time in Budapest vs Prague? The central tourist zone in Budapest is 10 to 15x greater in size than Prague.

Day 10 Depart

In April, Prague has Easter markets, the Netherlands has prime tulip season....so many good choices. The weather patterns are changing so quickly, so it's very difficult to predict rainfall patterns, so just come prepared. April is very much a transition month, based on our experiences.
Easter will add crowds the week before and after, so keep in mind which locations may be the most impacted.
We were just in the NL this past April, so I'll attach my Trip Report, perhaps it will have some helpful info. It's an incredible country to visit, but we did get the rain- and the wind. But next year could be more spring-like.
Enjoy your wonderful family vacation. Safe travels!

https://community.ricksteves.com/travel-forum/trip-reports/the-netherlands-belgium-april-2024-brought-the-rain-pants-needed-the-rain-pants