Three weeks!
I've lived in the US all my life and I haven't seen it all. I haven't even seen most of it.
If your trip is going to be sight-seeing and theme park-oriented, it's gotta be Orlando, especially if you have kids. Disneyworld, Universal Studios, Busch Gardens (rent a car), Sea World, Kennedy Space Center. Be aware that Florida can be unpleasantly hot and humid in the summer, especially for someone accustomed to a cooler climate.
It's hard to recommend things to see in the US without mentioning the Grand Canyon and the desert Southwest, but that's a long way from the east coast...about 2200 miles/3500 kilometers, and while the heat in Orlando in the summer can be unpleasant, the heat in Phoenix, for example, is better described as brutal. The train thing...I don't know. Europe has an extremely advanced rail transportation system. The US...not so much.
My guess, you'll get best use of your three weeks by staying in one general area and driving, or plan two or three destinations prepared to fly.