Grand Canyon–a must see! It’s just so incredible.
San Francisco–a must see! It’s absolutely unique, beautiful, fun, interesting. Take the trip to Alcatraz (book in advance). It’s fascinating and creepy and you get a boat ride, too! Also, take a walk over the Golden Gate bridge.
Las Vegas–I hear it’s quite the spectacle. I’m dying to go. The contrast between the natural Grand Canyon and the entirely man-made Las Vegas is supposed to be quite startling.
If you like hiking, there are plenty of great day trips to be done. You don’t need to carry overnight camping gear–just boots and a daypack.
For cheap hotels, I’ve had good luck with Motel 6. They’re cheap and clean and common. That’s about all they have going for them, but if you just need a place to lay your head at night, they’re great.
As far as whether you should book flights within the U.S.–just depends on your travel style. I, personally, like to explore one place thoroughly. If I were in Australia, however, I’d definitely make sure to see more than just one city. It sounds like if you did the California/Grand Canyon/Las Vegas thing, you’d cover a lot of ground, but won’t be too stretched out. (You could swing through Arizona on your way up to the Canyon. I love Arizona.)
You’ll notice that people haven’t recommended much to do in LA. I’ve never been there myself, but as far as I know, it’s not the most interesting tourist city. I suppose a Hollywood tour would be fun, right? Anyone? Anyway, it just depends on what you’re looking for.