I did that drive in 2005, so things could have changed, including my memory.
I went north out of Yellowstone north exit, following the river to I-90. I believe it's highway 89. I like that drive -- it's easy and safe.
I took I-90 west to highway 93, where I went north. Highway 93 goes along west side a pretty lake. I recall stopping for lunch at a Mexican restaurant right on lake (yeah, didn't expect that) where we had filled (stuffed?) churros for dessert. That I recall clearly.
From hwy 93 I took hwy 2 to west entrance to Glacier NP. All roads between the two parks were easy as best as I recall.
By the way, Glacier was a last-minute decision, so I hadn't researched much. I just selected roads on the map.
Returning I left Glacier to the east, where we crossed the Continental Divide through narrow roads in the mountains. Those were much more scenic and more to my taste, but a little harder to drive if in a larger motorhome.
I'd like to get to Glacier again because the main scenic route had not opened yet. At the time it was limited to vehicles 21-feet long if I recall correctly. Can't be too tall or wide either, so leaves most Class Cs out.