Welcome to the U.S. I hope you are enjoying Chicago. The city has many miles of bike paths you could ride to prepare for your September tour.
Your planned tour route from Glacier Park to Salt Lake City will be a wonderful tour. I don't think you could pick a more scenic route. I bicycled almost the same route (from Missula, Montana, to Salt Lake City in 1996 with two friends). We followed Adventure Cycling's TransAmerica route from Missoula, southeast to Yellowstone and Grand Teton parks. Sections 4 and 5 of Adventure Cycling's TransAmerica Bicycle Trail maps show that route and provide much useful information. From Grand Teton park, we generally followed highway US-89 to Salt Lake City. We averaged about 55 miles a day, taking 17 days for the tour, including some short-distance days for sight-seeing.
The route between Missoula and Dillon, Montana, includes about 4 mountain passes with long, but not extremely-steep, climbs, followed by many miles of downhill coasting. There are no significant mountains past Dillon. The entire route has spectacular scenery -- the beautiful Bitterroot Valley south of Missoula, "Big Sky" country en route to Yellowstone, the wonders of Yellowstone Park and grandure of the Teton Mountains, and wide open spaces in Utah.
The entire route has little traffic, even Yellowstone Park, since you will be there after the summer tourist season. Weather-wise, daytime temperatures should be pleasant and evenings cool, with frost likely. Be prepared for both warm and chilly riding weather. You can get average daily high and low temperatures by month for towns along the route from www.weather.com.
I have a journal (pre-internet, typed) of our tour which I could mail you if you're interested. Just let me know your mailing address.
If you do the tour, you will have a wonderful ride and see parts of the U.S. much different from Chicago and Illinois.