I may be misinterpreting this, but Amtrak doesn't serve Missoula. And in case you don't know, if you want to travel on Amtrak with your bikes, you need to find a station with checked baggage service. Shelby has it.
Missed that one. For ending around Glacier, the two most convenient Amtrak stations with baggage service is East Glacier and Whitefish. (East Glacier is only open during summer months.) Shelby is a bit further east and in the High Plains at that point, about 70 miles east of the park.
I wish Amtrak went through Missoula and also by Yellowstone. Hopefully they'll reinstate the North Coast Hiawatha line that went from Seattle to Chicago via southern Montana and North Dakota.
I have gone west to east twice. Both times I left from Seattle around the 25th of May. While, long the climb up the west side of the Cascades is not grueling in terms of overall steepness, and there is at least one section of downhill on the way to Rainy Pass and a short section after Rainy before you climb again to Washington Pass. As mentioned, Colonial Creek is a good jumping off point (and very pretty). If you stay at Newhalem you will enounter some good ups and dows before the climb starts in earnest, which it does right after Colonial Creek. From there, its about 32 miles to Washington.
I second camping at Colonial Creek. If I remember correctly, it was 6-7 hours of climbing from the campground to Washington Pass. There's a few campgrounds after the screaming descent down the east side.