OK. So, at a guess, the cheapest places for you to fly (unless you stumble across a deal) will likely be New York, Boston, Chicago, Washington, possibly Atlanta, possibly Miami. Of those, the U.S. Capital is really pretty darned cool (not literally, in August, unfortunately), and absolutely chock-a-bloc full of wonderful, free museums (even museums that appeal to children), and other cool stuff. The others are also lovely (I particularly like Boston), but to me at least stand out less (from one another) as "pick this one!" sites. And the capital isn't necessarily the right one, either -- but in some ways an obvious choice if your boys have not been to the US.
From those spots, you should be easily able to find "reasonable" tickets to major western destinations, and the US West really is strikingly lovely, very different from the UK (in a way that I think much of our east coast is not, in terms of geography).
You have two years (!!) to plan this. Why not engage your kids and plan a grand trip, defined in part by them ... they have to do the research and propose itineraries, or at least, destinations and bases for visiting them.
(As you must know but likely won't fully, fully appreciate until you are here unless you are the kind of person who can really "feel" a hypothetical budget ... gasoline is ABSURDLY cheap here (currently around $2.30 a gallon where I live, and even when it was "horrifically expensive" by our standards, it was $4.##). On the other hand, our vehicles are -- go figure -- less fuel efficient (but bigger, useful for a long trip and camping), and except in major metro areas and along the northern section of the east coast, non-car, non-plane transport is pretty impractical. Also our groceries and things are much cheaper, if I remember right. In a cheap but mainstream local grocery store, I'm currently paying around $3.50 for a gallon of milk, maybe $2.50 for a half-gallon of OJ, $2 for a dozen eggs, maybe $1.50 for a lb. of apples, $.50 for a lb. of bananas.)