Driving to Alpine, WY

We travel up the 89 from Brigham City through Alpine a couple of times a year in a 30' class A. It's a beautiful route and I highly recommend it as it winds up through Logan Canyon and down into Bear Lake. There are a couple of grades out of Brigham that will slow you down for a short while. There is another grade before beginning the drop down into Bear Lake but it's very doable. Just before you drop into Star Valley is a good grade first up then down but it's relatively short. The road through Logan is winding but the speed limit is 45 miles an hour with some passing lanes and big trucks are on it all the time with no problem.
All in all if you're not in a hurry and don't mind gearing down a couple of times it should be no problem. We just did the trip in mid May and will do it again mid-September. Our favorite part on the way to the Tetons.

Agree with wguss, it's a beautiful route, just take it easy.

Also agree with using route 89 through the Logan Canyon Scenic Byway up to Alpine. We have used that route several times on the way to the Tetons. At the beginning of the descent to Garden City there is a rest area on the right with an overlook of Bear lake, which is an amazing shade of blue on a sunny day.