Linda, Since you live in southern Michigan (as we did at one time), then you'll have to make you way through the Chicago area to get to I-80. Best to do this early Sunday morning. Stay on I-80 and pick up I-29 to I-90 to Rapid City, SD. Your first major stop could be in Hill City taking Hwy 16 from Rapid City. There are many campgrounds in that area and Rafter J Bar Ranch is a beauty but it's probably the most expensive. Others are Horse Thief and Crooked Creek and more. Using that as a base you could easily tour Mt. Rushmore, Crazy Horse and Custer State Park. You could also go a little further south to the town of Custer with more RV parks and the state park (driving through it even with only the truck is a must). If planning to stay in the state park, read the web site to avoid going through the tunnels which you don't want to do. Wind Cave Nat'l Park is also down that way.
Yellowstone - if you're really leery about secondary roads and mountain driving I'd suggest getting back on I-90 west into Montana and take Hwy 191 south near Bozeman to West Yellowstone. You'll have Gallatin & Madison Rivers for trout fishing - if you have time. :) It's an easy scenic drive. You could stay in West Yellowstone at RV parks and a few miles north of town is a favorite of ours, Baker's Hole, Gallatin Nat'l Forest c.g. with big sites and some have electric. It's on the Madison River - no reservations though but if you arrive early morning you'll get a site. You'd have to dump at a RV park in W. Yellowstone which folks do for a fee. Another option from W. Yellowstone would be to get a full hookup site inside the park at Yellowstone's Fishing Bridge campground. Folks will say it's a 'parking lot' but it's surrounded by forest and it's no worse than RV parks in W. Yellowstone. You'll be out touring the park all day anyway and will be ready for bed when you return. It's very convenient for touring the huge park. There are limited long sites so make reservations early. Madison is another nice campground inside the park that takes reservations but it doesn't have electric. It does have water fill and a dump station. W. Yellowstone has fuel and the park does, too. Expect to pay premium prices in the resort areas. Five nights to tour the park is plenty for the first time...you'll be back.
From wherever you stay to tour Yellowstone, definitely drive through the park on 191/287/89 to exit which will take you into Grand Teton Nat'l Park - a real beauty and altogether different from Yellowstone. It's like a miniature Alps. If you want hookups, Colter Bay RV Park is the only good place to stay and is a good base for touring the park. Take the gentle, guided, few-hour float trip down the Snake River and you might even see a moose or two in the water with you! This is the park for awesome hiking of all abilities. Stay at least three nights here or more. I'd advise not to try touring both Yellowstone and Grand Teton from one RV park. It's too much driving.
Then drive and exit Grand Teton N.P. toward Jackson, WY. It's a good place to re-stock groceries - a Smith's will be on your right after passing thru Jackson, along with a liquor store next door. There's also an Albertson's on the left before Smith's. You'll then come to Hoback Junction - a good place to fuel up - easy in/out and usually better price than Jackson. Then take 189/191 (easy drive) toward Rock Springs and I-80 east to Cheyenne and I-25 south to the Denver area and your destination there.
As Dick B stated above, get the "Mountain Directory West" online. It will give you mile by mile of roads that might concern you. If the hwy. isn't listed in that book then it's almost a sure bet that it will be an O.K. one to drive. Don't rely on a GPS mapping program solely, especially around Mt. Rushmore area, consult with good paper maps, too. Benchmark puts out an excellent state atlas for the western states with much detail, even campgrounds.
Also, for campgrounds check out the web site, 'RV Park Reviews' to get a general idea of what's out there. Until you get to the Mt. Rushmore area and then from Denver, home, I wouldn't worry about getting campground reservations. But do get them for the major stops if you're going in the summer. It will be very busy in those parks.
You're going to be passing many big and small towns and I really don't think you'll have a problem with fuel. Just fill up faster than normal - using the top half of your tank. As far as your pup - pets aren't allowed everywhere in national parks so do your homework on that.
Have a great trip. Yellowstone & Grand Teton are awesome parks!