We have used route 89 in the opposite direction from Brigham City to the Tetons several times. It is indeed a beautiful drive all the way. LittleJimsSon is correct about the grade up from Bear Lake. My recollection is that it is only about 2 miles long. Going back down to Logan is a much more shallow grade.
Assuming you will be driving through Yellowstone NP from West Yellowstone to Moran, you will already have some mountain driving experience going through the Park. If you decide not to take route 89 all the way, you can go right onto route 26 at Alpine Junction to Idaho Falls, then I15 on south. Do not be tempted to take route 22 west of Jackson to get to Idaho Falls, due to 10% grades & sharp curves.
Personally, I would use route 89 all the way.