I would recommend either coming later or earlier. Right after Labor Day is often very crowded with people thinking post-Labor Day equals no crowds. The water is also low and the sun high. Later in the fall the crowds begin to disperse and the sun angle is lower, making the fish less spooky. Late August is before the fall crowds come but after the summer crowds with kids leave.

The "best" fishing depends on what you're after in terms of technique, size vs numbers, and species. If I had to pick a first time to come west it'd be the latter two weeks of July. At this time of year the famous water will be crowded but the smaller streams, floatable water, and backcountry stuff is about as consistent as it ever gets, and it's when the dry fly fishing is best. For size, come from October 1-November 10 or March 5-April 15. Late March and April usually have comfortable enough weather (key word usually) and low crowds. Yellowstone Park doesn't open until Memorial Day weekend if that's in your cards, and most rivers are in runoff until at least July 1 (last year as late as July 25 for the Lamar, but the snow was heavy and spring late), with the Firehole, Madison, and Gibbon coming on earlier, in that order.

Regarding where to stay: Bozeman is not too far from everything except the Park (too far for a reasonable base camp for the park), but close to nothing. Gardiner and Livingston are right on the Yellowstone, Ennis and West Yellowstone on the Madison, and Big Sky on a good stretch of the Gallatin. If you want to fish more than one famous river (don't stick to the famous water!), you should plan on staying in different areas during different parts of your stay: West Yellowstone and Gardiner, for example.

The trip planner on my shop's website might help, though it's mostly focused on the Yellowstone and the park waters: www.parksflyshop.com/planner.htm

If you want to be taken into your backing, come the week before Labor Day and I can take you to the Missouri for carp averagaing 5-12lbs. Yes, I am serious... http://www.parksflyshop.com/carpfloats.htm