You'll have the best luck weatherwise flying early in the morning.
I've used PGA a few times for refuels and they've always been
cheaper than Airnav shows, especially from the Terminal, not
Classic. Although Classic has come out in the wee hours and
waived callout fees. DRO (Starbase)is also a reasonable
place to buy gas in that part of the world.

Your route skips Monument Valley, east of PGA, worth flying
through. Four Corners is just a slab with people standing on
it and lots of cars parked nearby. Mesa Verde is worth the
trip. Looks like you picked a fairly low route through the
Continental Divide, good idea, especially in a C172.
DO NOT forget to get a Grand Canyon SFAR chart.
As far as camping on field, I've been to a few airports that
claimed 'camping on field', and I'll be darned if I could
find the area they were talking about. Don't base your camping
on those airports with that description.
Personally I'd recommend Lake Powell/Monument Valley over
Bryce Canyon and even Grand Canyon, mainly because of the
restrictions. Also I hear Dauphin(e?) Island on the gulf in
Alabama is pretty cool.
Sounds like quite a trip, have a good time.
Chris