National parks, book asap !
As Buck said, by continuing to research you will find thousands of possible stops along the way and the forums and other pages like the Map centre above are full of ideas. Once you have listed places that appeal to you get a good paper map and sit down with the kids so you can all join in the fun, 'must See's only relate to the individuals on the trip. For instance, the National parks of Utah and Colorado would be of far more interest to me then Dallas and Memphis, but that's me. ;-)
You should be fine in Yellowstone come June but as Spring/summer can come late to the park and will only just be getting fully open if there has been a harsh winter, you could consider reversing your trip and get there later.
Is this RV yours or are you planning to rent ? I ask as you might be better off renting a 27/29 ft Class 'C' than a 40 ft class 'A' if you are planning on staying in campgrounds set in nature rather than resorts. Some do not cater for such big rigs and I would actually recommend you book places like Yellowstone and Grand canyon in advance and as soon as possible, even with a 30 ft RV numbers can be very limited and can book out within hours of the booking window opening. Check at nps.gov and select your park.
Once you are off Interstate your average mph drops considerably in a large vehicle and 300 miles [especially with kids] with a couple of short stops can add up to a day on the road, get in the mountains with an hour or 2 out the vehicle and half it again !