That's a Big "I Don't Know"
Welcome aboard the RoadTrip America Forums!
The answers to your specific questions are relatively straight-forward but others require a lot of "I don't know"s. You have the order correct as far as efficiency goes: Tetons, Yellowstone, Glacier, and the roads to connect them are include another national 'park' the Rockefeller Parkway between the Tetons and Yellowstone. From Yellowstone to Glacier, you would leave via either US-20 to the west picking up US-191 north at West Yellowstone, or US-89 north through Gardiner to Livingston. In either case, your goal is to get on I-90 westbound as far as Three Forks and there take US-287 north to its junction with US-89 at Choteau MT and then continuing on to the St Mary's Glacier Park Entrance and using Going to the Sun Road through the park. Upon leaving Glacier, the easiest route west is to just pick up US-2 to Bonners Ferry ID and then take US-95 down to I-90 at Coeur d'Alene.
As far as how much time you should spend in each park, that's a big "I don't know." It depends on how much time you have available for the entire trip and how you want to parcel it out among the various sites along the way, including those in the eastern US and those between Glacier and Seattle. Other big "I don't know"s include things to see and do along the way, accommodations, etc. Sure I could make recommendations, but those would be for places I liked. I have no idea of what your tastes run to. That's the beauty of a RoadTrip, you get to design it to be perfect for you. If you could give us some idea of what you like, then we could help you, but right now all we have is a blank canvas.
AZBuck