Nevada
You can also rent from Angel Air in Phoenix. They are quite relaxed about UK pilots - they have to be as they train the BA pilots in US flying before they let them loose in the 172's and 182's (hee hee !). Freinds have rented from a location in Scottsdale too but I don't recall the name.
The check out at Angel Air includes a high altitude/temperature element though not a full mountain course.
The trip to the Canyon in great and best done earlier rather than later in the day when the turbulence can increase. There is a specific chart for the canyon and shows the VFR routes for the main sights. Best tackled with a GPS.
I can also recommend the staff at Las Vegas north who could not be more helpful and even rustled up a car when all other attempts had failed and so made it possible for us to stay another day.
On that trip we went up to Monterey (jazz festival in September is good) and then came back down the coast - Big Bear in California is spectacular as is Catalina Island off LA before heading back to Phoenix.