To me Utah will be the most interesting part of this trip North, so, I say spend as many days there as you can.
Here's my whirlwind tour of some of the best places in Utah, by vehicle. Not a lot of time to actually explore, but, you'll get the flavor and this will wet your appetite to return for longer and longer visits. You're going to cover some long distances.
Leaving the 24th, spend that night in Valley of Fire.
Then on the 25th do a half day visit to Zion, as you probably won't find any parking to do even a short hike at the end of the Floor of the Valley Road, but, you might be able to drive it, unless they force you into a shuttle bus. If you get to see the main valley, one way of the other, great, if not, then just continue on Hwy 9, up the canyon wall, through the long cliff side tunnels and out the top / back side of the park. Just about ever time I got that way, it rains and that is fantastic the way the water runs off the rocks on top of the park. Same day turn north on Hwy 89, turn East on Hwy 12 to Bryce Canyon. If you haven't already got reservations at any of the Bryce Canyon campgrounds, and if it's late after noon or evening, then about 3 miles back west on Hwy 12 is Forest Service road 087, also known as E Fork Road, a well maintained gravel road. Go about 7 miles to Tropic Reservoir cross the Dam find the campground. Sometimes there are ATV's & motorcycles there, however, they are usually good about quiet times after dark.
On the 26th visit Bryce Canyon in the AM, go to the end of the paved road in the park, and then as you return, stop at the roadside turnouts to view over the edge, When you're done head on out East on Hwy 12. Escalante is huge and will deserve more time that you have on this trip, but, now you know where it is. After Escalante, Hwy 12 turns north and rides on Hell's Backbone Ridge. Amazing canyons on both side of the road. If it's been hot, Hwy 12 will climb into the Boulder Mountains were there can be some vacant camp sites along the route at some nice cool elevations. iI you don't find any campsites available, or you just want to continue on to Torrey.
I can recommend the folks at Thousand Lakes RV Resort, west of Torrey on Hwy 24 about 2-3 miles from the intersection of Hwy 12 & Hwy 24. You can tell them you are completely self contained (if you actually are) and that one of the out front "tent" sites will work for you, instead of the higher priced "RV" sites with 110 VAC. They have showers ! And, a swimming pool, too. We have never noticed any road noise. Re-provision in Torrey if needed (not much else in the next 100 miles or so). If you want continue East on Hwy 24 to Fruita, visit the Fruita campground or get a reservation to bypass the Thousand Lakes RV resort, or just visit on the 27th. You are now in Capitol Reef National Park.
On the 27th take the dirt road Scenic Drive going South out of Fruita. Enjoy the views in Capitol Reef N. P. both going in and coming back to Fruita. If you had more time Capitol Reef is huge with lots of dirt roads to explore. Continue East on Hwy 24, to Hanksville. EXTRA NOTE: If you had buckets of time, I'd say turn South on Hwy 95, the most beautiful highway ever paved (you'd see Hite Crossing of the Colorado, Natural Bridges and Comb Ridge) and go to Blanding. North on Hwy 191 from Blanding would take you to Moab, Arches National Park, Goose Neck, Canyonlands National Park. But, you don't have the time for this trip. Do it next time.
From Hanksville, you have about 40 miles to get to Hwy 50, and, you'll have to decide to turn West back towards I-15 (about 120 miles) and then North to Salt Lake City, or turn East towards Grand Junction in Colorado (about 120 miles). I've deposited you at the intersection of Hwy 24 and I-50, in the middle of nowhere. You've still got plenty of time to get farther North. You could even slow down and add another day or more in Utah.
I just got carried away while I had some free time to think about being in Utah. Those are my thoughts.
Pick what you want.
Have a great time.
