There should be no issues for you or your motorhome getting to the southern Utah national parks.
From Las Vegas take I-15 to Hwy 9 just north of St. George, UT. Take Hwy 9 east to Springdale and get a RV park there for touring Zion. The free national park shuttle makes stops in Springdale which is just outside the park entrance. Or you can stay in the park itself (highly recommended) at Watchman campground - electric sites and dump and water fill.
Leaving Zion, skip the tunnel because it is a steep climb and it doesn't sound like you want to do that. So head back west on Hwy 9. Then you can do one of two easy routes to Bryce. 1) go back to I-15 north and take Hwy 20 at Circleville east to Hwy 89. Go south on 89 through Panguitch and take Hwy 12 east to Bryce. Do not take Hwy 14 from I-15 at Cedar City. You won't like it. 2) Take Hwy 9 west to Hwy 59 at Hurricane. It will become Hwy 389 in Arizona. Follow it to Fredonia, AZ and head north on Hwy 89 and east on Hwy 12 to Bryce. From Fredonia you could also work in the North Rim of the Grand Canyon. At Bryce, Ruby's RV just outside the park gate is the most convenient and again, the free park shuttle stops there.
From Bryce to Capitol Reef you could take Hwy 12 east (our favorite) but you will have a steep hill near Capitol Reef. Many RVers do it and it's a beautiful drive through Escalante Nat'l Monument. Or...to avoid that hill, from Bryce take Hwy 12 west back to Hwy 89 north not far from Panguitch and pick up Hwy 62 to 24 east toward Torry and Capitol Reef. RV parks in Torry.
From Capitol Reef continue east on Hwy 24 to I-70 at Green River. Take I-70 east to Hwy 191 and Arches Nat'l Park. There are many RV parks in Moab. You can easily do Canyonlands Nat'l Park from Moab. Or you could stay at Dead Horse Point State Park at Canyonlands and tour Arches from there. Dead Horse is beautiful.
All of these routes are fairly easy. Stay as close as you can to these wonderful parks for the best overall experience. Plus, you'll save time and fuel. None of these routes have big mountains to climb.
Each national park has a great web site with lots of good information on camping, road issues/maintenance, things to do, hikes, etc. Check them out. Here is Zion's:
http://www.nps.gov/zion/index.htmHave a great trip! It's a favorite section of the country for us.
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel