September 8: Arrive in Jackson; drive the western route to Jenny Lake. Take the boat shuttle across Jenny Lake & hike up to Hidden Falls, Inspiration Point, or (if you want a longer hike) Cascade Canyon. Spend the night @ Signal Mountain Lodge or Jackson Lake Lodge?
September 9: Drive up to Yellowstone; Head toward Yellowstone Lake and then Hayden Valley. Wildlife viewing is great! Then continue up to Grand Canyon of the Yellowstone. Check out the overlooks along the rims. Walk down to the Brink of the Upper & Lower Falls as well as the Uncle Tom's Trail. Spend the night @ Canyon Village
September 10: Drive west to Norris Geyser Basin & explore that area. Take the Virginia Cascade Road and then, near Madison take the Firehole Canyon Drive. Continue west out of the park & explore what West Yellowstone has to offer. Perhaps stay at the Diamond P Ranch? It has chuck wagon dinners and horseback riding. A ride and dinner is $60.
September 11: Head back to Norris then up to Mammoth Hot Springs. Explore that area then head to Tower. Check out Wraith and Undine Falls, the Petrified Tree, and Tower Fall. Spend night @ Roosevelt Lodge?
September 12: Drive east and explore Lamar Valley. Then head south past Mount Washburn (hiking opportunity to top of Mt. Washburn) Once you get to Canyon, you've seen it all, so head west to Norris, over to Madison, and south to Lower and Midway Geyser Basins. Check out Fountain Paint Pot, Firehole Lake Drive, and take a hike to Fairy Falls. Spend night @ Old Faithful Lodge?
September 13: If you didn't hike to Fairy Falls the day before, do that first. Then explore the Old Faithful area and Upper Geyser Basin. Then exit the park and stay @ Signal Mountain Lodge or Jackson Lake Lodge
September 14: Float trip down the Snake River in Grand Teton NP!!! Spend night where you did September 13 or in Jackson
September 15: Whatever you can before you fly out
I think this itinerary might work out well for you. Hope this helps!