Hi. First, I just finished watching your team win the World Cup -- congratulations on that! Second, you sound like an organized guy, so these next tips are things which you may have already researched. Billings, Montana is not a big place so make sure your airline reservations are in order. If you need an alternative to Billings you can try Jackson (Jackson Hole), Wyoming, Great Falls or Missoula, Montana, Pocatello or Idaho Falls, Idaho. Make sure your rental car reservations are in order before you leave (again, Billings is not such a big place) and understand the terms of the contract. Most rental car companies only charge you per day and not per mile, but obviously you should make sure of the terms before you arrive. Expect the company to charge you a little extra to rent in one city (Billings) and drop off in a different city (San Francisco). Also, cheaper air fare and car rental arrangements are probably available in bigger nearby cities such as Boise, Idaho and Salt Lake -- although both are further from Yellowstone than Billings. Finally, have you considered renting a small RV instead of a car? That would nearly eliminate your lodging costs -- which are far, far more than gasoline costs.
A GPS is nice as an earlier post said, but personally I also like having road maps. There's just not enough detail on those big USA/Canada/Mexico road atlasses which show each US state on a different page. Get a road map for each state you drive through -- about 5 bucks each. You can buy them from Amazon beforehand or get them at any gas station as you drive for about the same price.
As an earlier post said -- Interstate highways are for getting places quickly but the back roads are always more scenic. Again, a good road map can tell you when are where you can take the scenic route. You should average 60-65 miles per hour on Interstates but figure 50-60 mph on smaller highways because some of them wind through mountainous terrain.
Your itinerary looks pretty good -- drive one day and sightsee the next day. One drive that might be too much is Day #4. You'll want to spend some time in the Grand Tetons and also in the town of Jackson, Wyoming at the southern end of the Tetons. All the roads out of Jackson are slow, winding, and spectacular. Figure on stopping short of SLC on Day #4 -- Logan, Utah, for example, is a nice town north of SLC with plenty of services. You can make up the travel time on Day #5.
Lodging could be an issue at Bryce Canyon since there is really only one big hotel by the canyon's entrance (Best Western). It has plenty of rooms for tourists but they charge heavily for the convenience of staying near the canyon. Alternative arrangements are readily available in the little town of Panguitch, Utah, about 30 miles north. I stayed at the Panguitch Inn on Main Street last April for $39 per night. The Best Western at Bryce Canyon wanted $120. Of course, if you have the RV then these issues mostly go away 😊
One day at the Grand Canyon seems a little rushed. The South Rim has more lodging but gets very crowded in summer time even though you can expect the heat to be like an oven. The North Rim has few accommodations but has a higher altitude, is cooler, harder for most people to get to and therefore less crowded.
Let me be the third one to say that the LA sidetrip seems out of place. You'll spend a lot of time driving into town and back out again. Consider driving directly from Death Valley to Yosemite (these are all back roads). You can get there by driving north through the town of Bishop, California and Mono Lake and then getting down to Yosemite from the east on the Tioga Pass highway (CA highway 120). Or, you can drive west from Death Valley and go over the Walker Pass (CA highway 178) to Bakersfield, California (or nearby) and drive north to Yosemite along the western slopes of the Sierra Nevada Mountains and the old gold rush country. If you take the southern route you can also stop at Sequoia National Park on your way to Yosemite. Sequoia is great for its great giant trees and for the fact that the summer crowds just aren't there. Yosemite is fantasic for its scenery but expect it to be crowded in August.
Hope you enjoy your trip.
Tim
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