Very popular, easy hike to a beautiful waterfall called Hidden Falls in Grand Teton National Park. If you just want to see the falls, there is a boat shuttle that will take you across Jenny Lake ($15 round trip). Or you can hike around Jenny Lake (about 4-5 miles round trip). One way shuttle tickets are also available.
A steady 1.5 mile (one way) uphill hike to the NW corner of Phelps Lake affords you the opportunity to make this 30ft jump into the rather chilly Phelps Lake with Death Canyon as the backdrop. Park and start hiking at the Death Canyon Trailhead!
Forest Service maintained natural hot spring pool tucked deep in Granite Canyon. About a 45 minute beautiful drive south of Jackson with the last 10 miles on gravel. $8 adult, $5 kid. Stop just before the hot springs and make the short walk down to Granite Falls! (and try to find the secret hot springs known as the Hippie Dip)
Rent a UTV from Scenic Safaries in Jackson or rent one from Turpin Meadow Ranch. They are great for getting deep into the National Forests and going fishing. They are street legal so you can drive them off the lot in Jackson. Leave from Turpin to gain quicker access to the National Forests.
Plenty of local shops to rent gear or do a guided trip with. Rivers to fish: Snake, Buffalo Fork, Hoback, Gros Ventre, Flat Creek, Spread Creek, Black Rock Creek, Pacific Creek. You will need to purchase a fishing license.