Ah - I didn’t notice that you were from Australia - so yeah, add an RV into the mix of driving on the “wrong” side of the road, and it’s a recipe for stressful driving.
If you’re sticking to main(ish) roads, you’ll never be that far from help. Obvious make sure you’ve got a decent cell phone. Some remote parts of the west may not have the greatest signal - when we were out in Arizona etc. in 2010, there were places where my cell phone’s nav application couldn’t get a signal at all. But unless you’re on really, really rural roads at really, really off times of the way, someone will be along.
It might be a while, though - so if you see a sign like “last gas for 40 miles” or some such, make sure your car’s tank is at least half full. I made the mistake of telling my husband “Oh, we’re fine” just as we passed such a sign… and THEN looking at the fuel gauge. We made it to the next town - barely. It was summer, in the desert, too. Fortunately it was night-time so we wouldn’t have cooked alive, but it was a foolish thing to do. Speaking of which, always make sure you have water and snacks when travelling in the more remote areas.
The one place where we were at all worried if something had happened was when we were leaving Zion NP to return to Kanab. The “more direct” route was undergoing major construction and the trip to the park had been difficult, so we left via a different entrance that was an hour longer. Unfortunately, by then it was quite dark, and we saw no other cars on the road. We were fine - but you don’t see roads that empty anywhere near where we live!
The rental company will certainly make good if the car fails. In the more remote areas, of course, it might take a while - and I don’t know how it would work if they had no office nearby.
Do make sure to reserve a car with unlimited mileage. That’s quite common nowadays, though I recently found when renting in a small town in Vermont, that anything larger than a sedan did NOT come with unlimited miles (renting the same larger car here at home still does). And, get a larger vehicle - an SUV or even a minivan. You can do the trip in a sedan (we did it in a Mazda 626, about the size of a Honda Accord) but especially with 3 of you, and 4-6 weeks’ worth of luggage, you’ll appreciate the extra legroom.
Consider getting the extra insurance that covers anything you might do to the car. For most Americans, it’s a scam - most of what it covers is already handled under your own auto insurance, but for someone from outside the country, it would reduce the headaches. Sadly, it is spendy - 15 bucks US a day is not uncommon.
Time of year: You won’t want to do it in the wintertime. Some of the highways in the midwest and west can get shut down in a bad snowstorm - even the interstate highways. If you do travel that time of year, make plans to stop if it looks dodgy. Snow is NOT to be underestimated, especially if you’ve never driven in it - and ice is even worse. High summer would be miserable in the southern parts, though air conditioning helps a lot. When we travelled through Las Vegas in July 2010, it was 115+ F (45ish C) - but the car’s air conditioner made it pretty tolerable when we were actually driving.
You could do a hybrid trip - part train, part car. The downside there is that you’d pay more for the car as you’d be dropping it off in a different city.
Good parts to skip over (i.e. take the train instead of driving): pretty much all of the east coast between Jacksonville, FL and New York City. Well, there are certainly places to see en route (e.g. Washington, DC; Baltimore MD has a rather nice aquarium etc.) but nothing unmissable. Ditto from New York to Chicago. I can’t speak for much of the southern route - probably nothing unmissable between New Orleans and Jacksonville but I haven’t been that way, so don’t take my word for it.
You really don’t want to have a car in New York City: if you’re driving, you go to the hotel the first day, have them valet-park it, and leave it until you’re ready to leave town. I’ve spent a lot of time in Manhattan and the transit is so much, much easier (assuming you don’t have luggage) than driving.