Some help to get you started.
Hi, and Welcome to the Great American Roadtrip Forum.
I know there is a lot of romance about travelling by camper or RV, but it is nearly always cheaper to go hotel and car. Remember, in the US you will not be able to travel the distances you do at home, without coming to towns and cities and accommodation.
Driving from Perth to Melbourne is almost equivalent to LA to the east coast, but don't imagine that it can be done in four days. That is a very hard five day drive, in the car for ten or eleven hours. without any sightseeing. Six days makes it a little more relaxed and a week makes it a really nice trip.... but that is assuming you are not stopping at any of the thousands - nay millions - of attractions along the way. You will not anywhere find the open spaces you have across the Nullabor.
Three months would certainly give you the opportunity to see many, if not all, the major attractions, with time to actually see them, and for your children to enjoy and benefit from their visit.
Regardless of which way you travel, start thinking loop trip. It is very expensive to pick up a vehicle in one city and drop it in another. Cross country could run into four figures. Secondly it is always cheaper to fly in and out of the same airport.
Do you have any good large paper maps of the USA? One you can put up on the wall where you will see it every day? My preference is for the National Geographic map available at Map World. The larger one is excellent for the detail it gives. Furthermore I suggest you get a copy of the Rand McNally road atlas which has detailed maps of all States with scenic routes and attractions marked as well as all cities and towns and roads and routes. Both are invaluable to give you some idea what there is and where it all is in relation to one another.
Use post it notes on the wall map to mark attractions as you decide which ones are on your must see list. Once you have some spots on the map, the members here will be able to help with routing, distances and times, etc. (One warning, ignore computer mapping travel times. They are pure fantasy and do not relate to real world conditions.)
Lifey
Three Months is More than Enough
When I did my first 'Once Around the Country' RoadTrip, I had three weeks and did it in an AMC Gremlin (AMI Rambler Gremlin in Australia). I got to see far more than I would have imagined and still had time to visit friends along the way. Three months and an RV are luxuries I could only imagine. So yes, it is enough time to see an awful lot of the US at relatively relaxed pace and get back to your starting point without once retracing your steps. I wholeheartedly agree with Lifey that you ought to close the loop on this. Leaving any vehicle, but especially an RV, thousands of miles from where you pick it up is going to get very, and needlessly, expensive. Whether to use an RV at all is another question and whether the additional expense and added hassles (urban driving, finding parking, poor fuel mileage, etc.) are worth it is entirely up to you.
You're probably looking at around 10,000 miles to complete a circuit of the US that would get you to roughly every region and many of our most scenic/historic/cultural sites so you'd only need to drive about 150 miles a day or less. That is eminently doable and leaves you a great deal of time to actually explore and enjoy whichever sites you choose to visit.
AZBuck