No, I don't think you'll have much use for the snowshoes. I'm in Montana an hour north of Yellowstone right now, and today they're calling for snow down to about 7000 feet elevation, which would put snow in much of Yellowstone Park. They were calling for 6-12 inches at Cooke City on the northeast edge of the park. But this the first snowfall we'll get, and much of it will probably melt in the next few days as the temperatures are expected to rise into the 60s and 70s by the weekend. Usually we get a nice snowfall sometime about now, but hardly ever in early September.
I'm very partial to Montana rather than Wyoming, mainly because Montana has a lot more enlightened stream access laws. Both states have beautiful rivers and great stream trout fishing, but you're not limited much in Montana in where you can float and fish and get out of the boat, or wade and fish for that matter. Much of the wading size water in Wyoming is private, same as Colorado.
So if it were me, I'd just sightsee much of Colorado and Wyoming, headed for Jackson and the Tetons. A guided trip down the Snake through Teton NP is well worth your while, absolutely gorgeous scenery and nice fishing for Snake River fine-spotted cutthroats. Then I'd head up into Yellowstone. Warning...it's crowded, even in early September, these days. We were in the park yesterday and the number of people was a little dismaying. We went up into the Lamar River Valley, and there were a LOT of fly fishermen fishing the Lamar, in some places lined up along the banks.
If you want to see all the thermal features in the park, just spend a day driving the roads of the park. If you want to view wildlife, spend some time in Grand Teton on the various roads, where you are likely to see moose, bear, and bison, then head quickly up through Yellowstone from the Tetons to Fishing Bridge because there isn't a lot of good wildlife viewing in the southern part of Yellowstone. There's a loop starting at Fishing Bridge junction that goes around to the Madison Junction, then to Mammoth Hot Springs, then to Tower, and then over the Dunraven Pass and back to Fishing Bridge. That's where you'll see a lot of wildlife. Then retake the part of the loop over the pass to Tower, and out the Lamar Valley, where the rest of the good wildlife viewing is. For a terrific scenic drive, plan it so that once you head out the Lamar, keep going through Cooke City and over the Beartooth Highway down to Red Lodge and then I-90 in Montana.
Then to get in any more good guided floating and fishing, head west a bit on I-90 to Livingston, where we live half the year, and the Yellowstone River. Lots of guides, lots of beautiful water on the Yellowstone, and also good wading water on the Boulder and Gallatin, and more floating and good fishing on the Madison, all within an hour of Livingston.
As you can see, there's a lot to do and see, so you have to make some plans and set some priorities. As far as I'm concerned, this area is the best part of the West.