Utah highways are excellent for a big rig.
You can do all this in reverse but one route would be I-15 south to I-40 east to Williams, AZ. Stay there and visit Grand Canyon. Or... if you can get one of the big sites in the park itself, Trailer Village is a great place to stay. The free shuttle stops right there and you can even walk to the Rim.
Leaving the Canyon ... if staying in Williams go east on I-40 to Flagstaff, AZ and pick up Hwy 89 north. Or... if staying at Trailer Village in the park just head east out of the park to Hwy 89 north.
Take 89 to Page, AZ (Glenn Canyon Nat'l Rec Area) and on to Kanab, UT - Coral Pink Sands State Park is a nice day trip and also Best Friends Animal Sanctuary - awesome. Sign up beforehand for a in-depth tour or do minimal on your own. Grand Canyon North Rim is also easily accessible from Kanab/Jacob Lake & Hwy 167 to the Canyon. It's the 'quiet' side of the Canyon. Stay in Jacob Lake or Kanab.
If you didn't want to go to Page or Kanab then from the South Rim and 89 you could pick up 89A to Jacob Lake and Fredonia.
Either way... get to Fredonia, AZ if you don't want to do the Zion Tunnels.
From Fredonia, AZ take 389 west which becomes UT 59 taking you to Hurricane, UT and Hwy 9. Go east on Hwy 9 and there are RV parks to stay for visiting Zion National Park. You can exit Zion via the tunnels or backtrack on Hwy 9 to I-15 north. Pick up Hwy 20 east taking you to 89. (Don't use Hwy 14 - it's steep).
From UT 89 again you can pick up Hwy 12 east. You'll have one tunnel but it's tall enough for the biggest RVs and semi trucks and it's short and flat ground. Soon after will be Bryce Nat'l Park. Ruby's is a nice place to stay and the free shuttle stops there. If able, be sure to do a hike down into Bryce Canyon. It's a fairy wonderland down there!
Leaving Bryce continue east on Hwy 12 and you'll come to Kodachrome State Park if that's a destination or else do it as a day trip from Bryce. You could also make a stop on 12 at Escalante and explore along Hwy 12.
Continue on Hwy 12 to Torrey, UT for visiting Capitol Reef Nat'l Park.
Then take 24 to I-70 or for a beautiful drive take 24 to 95 toward Blanding, UT. From Blanding you could take UT191 south to 163 to Monument Valley for a tour. Stay at Gouldings RV. Then north on 191 to get to Moab.
If taking 24 to I-70 go east on I-70 to 191 south for Moab/Arches/Canyonlands Nat'l Parks. Definitely stop at the Moab Visitor Center for some trail maps to take your Jeep for wonderful riding. There's a nice evening Colorado River cruise at Moab. Arches is a hiking park and if able make arrangements beforehand to do the Fiery Furnace ranger led tour - very unique. Delicate Arch is nice late afternoon/early evening - especially if there's a full moon. Take advantage of the ranger programs in the national parks.
Each national park has a great website giving things to do, hikes, places to stay, road information and any special alerts. Check them out. Here is Arches:
Hope this helps! They are all favorite places of ours.