Sun, 7/28: fly to Bismarck, ND, drive 2 hours to Dickinson
Mon, 7/29: spend the day exploring Theodore Roosevelt NP (South Unit)
Tue, 7/30: leave early, drive 5 hours to Wind Cave NP, do ranger-led cave tour at about 3pm, drive to Mount Rushmore, spend night in Rapid City
Wed, 7/31: Badlands NP, night in Rapid City
Thu, 8/1: drive 6 hours to Denver. Spend a week and a half in the area doing all the things, including Rocky Mountain NP. If you want to do a really long day trip, include Great Sand Dunes, NP, but for 7.5 hours of round trip driving, it's not *that* much reward.
Tue, 8/13: Tisha B'Av
Wed, 8/14: Drive 5 hours to Black Canyon of the Gunnison NP. Spend maybe 2 hours there? Spend the night in the area, or on the way to the next NP.
Thurs, 8/15: Drive up to 5 hours to Mesa Verde NP, spend a few hours in the NP, continue on 3 more hours to Gallup, NM for the night
Fri, 8/16: early start. drive an hour to Petrified Forest NP, spend an hour or two, drive another 2 hours to Flagstaff for Shabbos.
Sun, 8/18: drive an hour and a half to Grand Canyon NP (can drive motzaei Shabbos), spend night (or 2) Grand Canyon Village
Mon, 8/19: Grand Canyon NP, head to Page, AZ for the night, 2.5 hours away
Tue, 8/20: drive 2:15 to Zion NP (probably skip Antelope Canyon), spend the day, night in Springdale?
Wed, 8/21: drive 2 hours to Bryce Canyon NP, spend day, then drive 2:30 to Capitol Reef NP, spend night there
Thu, 8/22: early day, Capitol Reef NP for a 3(?) hours, drive 2 hours to Canyonlands NP, see that, night in Moab
Fri, 8/23: EARLY sunrise at Delicate Arch in Arches NP, another hour in Arches, then 4 hours to Salt Lake City for Shabbos
Sun, 8/24: 5 hours to Grand Teton NP, spend night nearby
Mon, 8/25: Yellowstone NP, 2 more nights
Wed, 8/26: 10 hours to Glacier NP
Thu, 8/27: Glacier NP
Fri, 8/28: fly home from Glacier Park Airport
17 National Parks, Shabbos and 9 Av with minyan.
Highly recommend RV/tent for at least part of it. Your previous comment makes way more sense.