Reading these posts from the UK, makes me so happy to live and fly in the U.S. Southwest. It's great. I agree about at least a C172. Agree about staying out of SOCAL, So much to see a further east.
Las Vegas, use one of the relievers(vgt), LAS is expensive. Grand Canyon, be careful about SFAR area or better yet grab a helicopter trip. Monument valley (picture in a previous post). Sedona, watch winds at both ends (be high over end).
Many, many more--have fun