With Grand Teton National Park, and Yellowstone National Park both in the upper Northwest corner of Wyoming, the area has a lot going for it. In addition to the tourist attractions of the national parks, the area is famous for it's alpine rock climbing, and white water rafting in the summers, and skiing in the winter. Jackson proper is a small town, but very touristy and expensive.
Free camping options are available in the National Forest surrounding the parks, but if you plan on camping in the parks it is best to get a reservation in advance. Though a 30-45 minute drive from Jackson, the Snake River Canyon, between Alpine and Hoback Junction, has several campgrounds that offer decent camping, and are generally less full than the camping areas in the parks.
One final note, be aware that this is bear country. Store all food in bear resistant containers (and no, a cooler is NOT a bear resistant container), and/or hang it when in the back-country. Campgrounds typically provide a large metal box for putting food in while in the campground. Although bears stealing food is potentially dangerous and deadly to humans, it is also extremely dangerous for the bears. Bears that become habituated to stealing from campers may be relocated or even euthanised.