To the OP:
I lived in Jackson, WY for 4 years and I have car camped, backpacked, and horsepacked extensively in the area. I have not been everywhere, but I got to see a little bit. For the diapers, treat the inserts like toilet paper and you will be good. Leave no trace is up to you. Most of the USFS campgrounds in the area WILL NOT have trash cans, unless they are high traffic areas along a highway (the Hoback River for example). The National Parks will have trash cans at almost all locations. For camping, I agree with the 100 yard principal although often I make it shorter depending on the space. I think the key is don't sleep right on top of your cooking area or your food storage. Be sensible. If you hang a bear bag, I recommend tying your pots to it or a bell so when the bear comes and starts shaking the tree or pulling on your rope you will know in the night and you can try to scare it away. Bring bear spray...much more effective than a gun -- read the BYU article.
Some highlight spots...
National Parks -- Teton is much prettier than Yellowstone. The Jenny Lake Loop road is great for the views at sunset. Guaranteed elk sighting -- go to the Lupine Meadows trailhead in Teton around sunset or dawn. Pretty much a guarantee. Want to see a Moose -- drive the Gros Ventre river road really slowly. Bison? Head out the Gros Ventre river road and drive around by Kelly, Wyoming and toward Antelope Flats...bring binoculars and scout the sagebrush. If you are in that area, take the dirt road that runs north and south to Mormon Row to see the old barn that is in pretty much every sunset picture of the Tetons. Go at sunset and make your own scenic photograph with your family. Look around Antelope Flats for antelope as well...usually pretty easy to spot. Great camping up the Gros Ventre River valley. Look for a place on the map called Soda Lake if you have 4wd. Good lake for a canoe and it can be decent fishing.
Yellowstone, see the usual suspects...I think the Grand Canyon of the Yellowstone is better than Old Faithful. Head out to the Lamar Valley for great wildlife.
I won't give you my secrets for the best places in that area which are all outside of the National Parks, but I will say that all of the surrounding National Forest is just as good or better than either of the parks. If it were me and I had two weeks, here is my driving itinerary.
Come in from the south, head to Pinedale, Wyoming to see the Winds and that beautiful headwaters of the Green River. From there, drop over the Jackson and take it all in. Head up through Teton and Yellowstone over several days...I think 4-5 days for that area is reasonable. From Yellowstone, drive out the NE entrance and either head over Beartooth Pass to Red Lodge, MT or on the Sunlight Basin Road toward Cody. Both are amazing scenic drives with abundant camping and fishing. Drop down into Cody and spend a day or two. From Cody, head back over the Big Horn mountains and stop to camp and fish and then from there head east back toward St. Louis.
But hey, there are so many options! Grab some good maps and pick pretty much anywhere in that area and it is beautiful.
