I agree with pivobar on both counts:
- Interesting places will start in Western Montana
- You'll really need to do detailed research, to find out how you can effectively operate as a pdestrian once you disembark.
Are you willing to rent a car at all? Are you expecting to travel very light (recommended)? I could recommend East Glacier, Essex, West Glacier, Whitefish, and the Columbia River Gorge ... but I'm reluctant to do so if you're going to be stuck and immobilised.
I think it's safer to recommend Missoula (on the line or nearby?), Sandpoint, Spokane ... these are big enough and interesting enough to warrant a day, and being a pedestrian may not be a total disadvantage.
But overall, AMTRAK journeys are to my mind best done continuously as a destination in itself, rather than being seen as hop on hop off transport services. The lack of a car, plus carrying luggage, and the spread-out nature of just about every place west of the Mississippi, makes your goals rather difficult.