There's a lot of good info in LfC's article, definitely check it out.
It's a big park so where you're going to be staying will dictate to a degree where you will fish. Camping or not?
Yellowstone River, Lamar River, Soda Butte Creek, Slough Creek, Gardner River, Gallatin and Madison all good this time of year though the Madison will get better later in September and early October. 5,6 or 7wt will do the trick though I'd go with a 6 or 7 on the bigger waters especially if there's much wind.
I'm not a technical fly kinda guy so I'll give it to you in laymens terms. You're gonna hit prime hopper time, take plenty of hoppers and other terrestrials. Yellowstone is primarily caddis country, so the elk hair and all the rest of the caddis dries are essential. I've had good luck in some areas in September with the mahogany dun. When it was working, seemed like nothing else would.
There are some great books on fishing Yellowstone, one of my favorites is "Fishing Yellowstone Waters" by Charles Brooks. It's a little dated but some things don't change. A lot of great info.
It might pay to double check the reg.s for the rivers you plan to fish, as many rivers in that area and other areas of MT are closed due to high water temp (i.e., the yellowstone is closed on almost it's entire length from Noon to Midnight, as is the Bitterroot, and Clark Fork. I think the Big hole and Gallatin are closed completely, but double check the FWP for correct info). Restrictions are due to lift on Sept. 15th if temps drop. Not to preach, but one person can kill a lot of fish when the water is as hot as most rivers are down there now, no matter how quickly played or cautiously handled.
Last time such broad closures were implemented, the payoff was incredible. When the temps dropped and the rivers reopened, the fishing was awesome. Lots of hungry fish eager to take flies.:thumb: