First week in September? Realistically, anything you like in Montana- you just have to acknowledge that the weather might be unpredictable, bring decent raingear, some plastic bags for your feet, etc. But odds are in the first week you'd be fine.
Regarding Montana, though, I'm working totally second-hand, since I haven't hiked there. (The guy to ask for good ideas in that region is Dave Chenault.) But from what I understand:
2-3 days probably isn't enough to do the big loops in Glacier, unless you're a sock puppet for Andy Skurka. There are many shorter out-and-back hikes there, though. Just get a map of the park and look for them. The northern half of the park is the more classically alpine, but the southern half is less crowded and still has world-class scenery. Not that there will be ridiculous crowds in September. There is an at-large camping area around the Coal Creek basin, which I mention because I personally find reserved campsites and rigid itineraries to be annoying.
You could probably make it to the Chinese Wall in the Bob and back, though that would be pushing it timewise, again.
In Utah, the first week in September might still be a bit early in the season, i.e. ridiculous hot. So stay high. On the plus side it is (IIRC?) probably out of the monsoon season so less risk of flash floods, yet there should be water available at the usual sources from the recent rains. As I recall October is when the leaves change color, and a very popular time in the region.
(As fate would have it, I'm currently researching Utah hikes for May.)
The usual national parks and monuments are of course options- Zion, Capitol Reef, Canyonlands, Grand Staircase, etc. Canyonlands can be annoying because water sources are not common in the park. Zion generally has decent water sources. I cannot speak intelligently about the others. Zion also has an online campsite reservation system that's pretty handy, actually. They open them in 3-month blocks, so for instance right now March, April, and May are open to make reservations. On April 1st it will be April, May, and June. Etc. You really have to jump on them on the first of the month to get what you want in peak season, which is spring and fall. But they reserve half of their campsites for walk-ins, too.
What I'm currently looking at are more off-the-beaten-path stuff, like Dark Canyon, Coyote Gulch, and Grand Gulch in the southeast corner. (All mentioned to me by people more informed about the region than I am.)
Grand Gulch is apparently lousy with Anasazi ruins and might be a great idea for 2-3 days, particularly Bullet Canyon. It requires a permit for overnight camping, available at the Kane Gulch ranger station. You can reserve ahead of time or walk-in. The Trails Illustrated map for Grand Gulch seems pretty good- it even has all of the ruins, arches, and water sources numbered.
I'm pretty sure that Coyote Gulch also requires a permit, since it's technically in Grand Staircase NM and Glen Canyon NRA. Dark Canyon is a wilderness- no permit needed.
For your 2-3 day itinierary, the region is lousy with shorter but interesting non-technical canyons. The accepted source is Kelsey's book:
http://www.amazon.com/Non-Technical-Canyon-Hiking-Colorado-Plateau/dp/0944510272/ref=sr_1_1?ie=UTF8&qid=1426604228&sr=8-1&keywords=Kelsey+non-technical
He tends not to talk an awful lot about water sources, though, so some internet searching may be needed to feel secure on that issue.