If you heading from Yosemite to Las Vegas (or as some call it --'Lost Wages'), I suggest swinging through Death Valley along the way. The coast north of SF has the Redwoods and the world's tallest trees (100m+). Inland from the coast, NE California, has Lassen Volcanic NP and Lava Beds National Monument. Highway One, south of SF in the Big Sur area, is a world class scenic drive, and the road is twisted and narrow. Drivable in an RV, but you may not enjoy the experience. The Oregon coast is RV friendly, and very scenic. Southern Washington has Mt. St. Helens NM which erupted in 1980. Seeing how fast the area recovered from a barren moonscape to a young forest again is interesting. NW Washington has Olympic NP and a temperate conifer rain forest.
If you are heading between Bruce Canyon and Yellowstone you can stop at Arches NP, and Dinosaur National Monument. Several less popular smaller state and county parks in Utah may be worth a stop if you have time. You can spend two weeks at Yellowstone/Teton areas and still not see it all. Zion NP is close to Bryce, but you may find it is a lot like Ayers Rock -- big sandstone. Not too far off your route is Mesa Verde NP in Colorado and its cliff dwellings (apartment complexes, small self contained cities actually, housing several hundreds to a few thousand people, built between 900 and 1300 AD without wheels or metal). I am trying to think of sights you would not see in Australia. You are travelling all this way, so I would not want to steer you to something similar to what you can see at home.
As a general rule, you can free camp anywhere on National Forest and BLM public lands unless specifically prohibited in a local area. Conversely, National Parks do not allow free camping, restricted to designated spots only. Most National Parks are surrounded by National Forests and BLM lands, so you can free camp outside the NP, and drive in each day to see the sights. A bit of a bother that way, but it is cheaper (no campground fees and no campground regulations).
Depending on the winter, the higher mountain roads may not be open until June. June is when the SW starts heating up and Death Valley could easily hit 40c+ in June (the record in Death Valley is 56c). The coast will be fine any of the months.