My advice if going to echo the others, rent a circa 25 ft motorhome, this should be big enough to be comfortable for your family of 4, particularly considering you have smaller / younger kids that are just old enough to be semi-independent (can bathe themselves, etc.). A 25 ft coach can fit in and go lots of places that are hard to reach in a 30 ft coach, if you were buying or even if you were American and renting from your home area I might say go bigger to have room for your belongings, but since you will likely be limited in how much stuff you are bringing due to airline baggage limits this is less of an issue, ie no bicycles, BBQ grill, etc. We have a 28 ft class A motorhome, and while we can fit in most places we want to go, I will admit I sometimes envy the ability of those 23-25 ft class C's to basically fit in a standard parking space (you can see our coach in our avatar photo, backed into a standard parking space and overhanging the curb at the rear).
As to where to go, to see the most in 6 weeks, in the summer a lot depends on if you want cultural experience, scenic vistas, vast open spaces, a variety, etc. Assuming your dates of travel are limited to July-August, be aware much of the southwest US will be VERY hot, as in the kind of HOT you don't get in the UK. I have been in both Las Vegas, NV and Phoenix, AZ in July-August multiple times in the past where the daytime highs were around 115-119 (46-48C) degrees F, with lows after midnight still being around 100F (38C), sure it is a dry heat, but that is still very hot. The south is not fun either in July - August, I live in Louisiana and our July-August daily highs are in the 95-100F (35-38C) range most days, but is also miserable as we typically have 75-95% humidity.
This leaves the question where is the pleasant weather in the US in July-August, the answer is the extreme north eastern, New England states, the western mountain states, and the Pacific coastal areas. I would not advise either New England (lots of narrow roads) or the Pacific coast (lots of people,) for first time motorhome drivers.
This leaves the mountain states, and pacific northwest which potentially include plenty to see and do in 6 weeks, I personally would consider renting in Denver or possibly Salt Lake City and making a loop ending back up where started. The 2 big RV rental outfits in the US are Cruise America and El Monte, they both have similar offerings, which are basic class C RV's which are custom built for the rental market, and are rather utilitarian (less stuff for the renters to break) with no TV, no retractable steps, no awning canopy, .... There are also some other companies like B&B rv in Denver that rent more standard RV's (I know nothing about B&B other than seeing some of their units in campgrounds and speaking with a few of the people who rented them that seems to have positive things to say).
The problem with the mountain states, and for that matter much of the western US is that everything is so spread out, which means to see much you have to often drive a good ways. To put this vast amount of space into perspective for you, it is possible to drive from Amarillo, Texas roughly due north to Rapid City, South Dakota a distance of nearly 700 miles without ever passing within 50 miles of a city of over 10,000 people, and only passing through a couple of towns of over 2,000. Or just take the state of Wyoming alone, which according to google has 3% more land area than the UK, but only has a population of 578,000 people which is less than 1% of the UK population.
What I would suggest doing would be something like, fly to Denver (Denver is known as the mile high city due to its elevation at 5280 feet, so tends to be cool even in summer), it is located on the vast flat lands just east of the Rocky Mountain range which runs roughly north/south across the western US. Don't plan on going anyplace east of Denver, unless you want to see vast areas of mostly flat land much of it farmland for 500-1000 miles in any direction. Take a few days and experience Denver, the culture, either using a rental car, or simply use Uber, etc. then pick up the motorhome, and head out. I would probably go north to see Wyoming, maybe Montana go west, perhaps to Idaho, south to Utah, and back across I-70 to Denver. There are numerous national parks, many of which are in Utah, but those may be a bit hot that time of year, there is also plenty to see outside of the major national parks. There are also plenty of options you might consider taking a southerly route out of denver and looping to the west, though this may be a bit warmer, particularly if you get down to Arizona and parts of Utah.
p.s. good arguments could be made for starting in Denver, Salt Lake City, even Las Vegas or some other regional cities depending on what you want to see.