Oh no, I just wrote a whole post and somehow erased it. I will try to recap. I'm American and have traveled a fair bit around the country. Unless you really want to just do super fast tourist highlights, you will need to confine your trip to one area of the country. Train travel is a joke outside of the Boston-NYC-Washington, DC corridor. The southeast is BRUTALLY hot and humid in the summer, which I hate, so personally I wouldn't go south of Delaware unless I was going to spend 99% of the time at the beach. Even Disney World is awful in the oppressive heat of July and August. There are beautiful beaches all up and down the east coast, though. My favorite beaches New England, from the rocky cliffs of Maine to the family-friendly villages of Martha's Vineyard. The water is cold though.
The US is so large and so diverse that it's hard to recommend a specific itinerary. If you are city people and you want to hit as many famous places as possible you could do Boston, NYC, and Washington, supplemented by relaxation time in Maine or the Outer Banks of North Carolina or even the Jersey shore, which is quite lovely in many places. If you want to see jaw-dropping scenery that is truly different to what you've probably seen in Europe, the American West is amazing: Colorado, Wyoming, and Montana are all fantastic in the summer, though it is BIG country so you will spend a lot of time in the car. Texas, Arizona,andNevada are also quite hot the though people claim it's a "dry heat" so not as bad as the southeast. A road trip down the coast of California alone would be a great 3-week trip, though it may be further than you want to fly or drive. Public transport is very limited outside of big cities, even in the close-in suburbs, so you will likely need to hire a car now and then unless you just stick to NYC, San Fran, etc.
Outside of really major cities like NY and San Fran, hotel rooms will be larger than you are probably used to seeing in Europe, though still tight for 2 adults and 2 teens. I highly recommend Airbnb or vrbo to find a nice place that will fit your family. You can usually book them for just a couple of days, although homes in popular beach resort areas may require a longer time.
There are many more possible options. I would just caution you to watch the heat, if that bothers you, and to try to avoid the temptation to see too much at once. You may find it much more enjoyable to slow down and get to know just a couple of places. The oppressive heat of a Washington summers is hard to even describe.
Feel free to PM me if I can help you further. Good luck!