What we don't know is why you selected this route, which is not the very quickest way to get to the Grand Canyon. Having driven from the Grand Canyon to Prescott, then on to Palm Desert then on to L.A., though not in one day, I didn't find it the most scenic route. We did enjoy Prescott. I'd rather get to the Grand Canyon as quickly as possible, and have more time at the GC, but each person's tastes are different.

The lodges at the canyon's rim are pricey, but, IMHO, are most worth the cost if your stay will be brief. Not all of the lodges in the park are on the rim. We stayed at the Kachina Lodge, and could see the canyon from our room. Williams and Kingman are not bad choices. Much depends on your ability and desire to do a lot of driving in one day. If you think you can drive as far as Flagstaff, it's not a bad choice, with a wide variety of lodging, and you could drive into GC Park and on to Vegas in one long day. You wouldn't have all that much time in the park, but if it's the best you can do, so be it. If you can push on a bit further, Tusayan is just outside the park boundary. That would at least give you more time in the park the next day.

If you are doing this in the summer, I'd even more strongly suggest just getting to the higher elevations of the Grand Canyon asap the first day, via I-40, rather than going via I-10, etc. In that case, staying in Williams (we stayed at the Days Inn) or Tusayan or inside the park would make sense. Yes, you would be backtracking some of the same route (I-40) on your way to Vegas, but you could always take Hwy. 66 (passing Grand Canyon Caverns) rather than I-40 part of the way.