Be sure to check up with some Aussies that have done the trip. A lot have come over here and done the trip we met one at a KOA in Kingsport Tennessee a couple years ago. He was over as an exchange teacher program and was traveling around the country before heading back to Australia. I'm sure many of them can give you better advice than we can. Lol
You will probably want to through in an rv park after a few days of camping in national parks and etc. especially if you are going without any hook ups. A couple of people mentioned KOA. While they can get a little on the high side, they cater more towards families and kids, with lots of kids activities.
Keep us all updated on your itinerary and I'm sure many of us would enjoy meeting up with you on your journeys.