Go to the nearby University and use their Library to do your research. Visit the flood museum or the World war 2 Museum (Legacy Museum). Go to the City parks and walk several laps around the park.
There are lots of fine dining or there are 3 big grocery stores where can provide for your own eating.
Don't forget to take a hike up the Rexburg Butte, which is an extinct Volcano - (tough hike)
Maybe you would like to go out to the Sand Dunes and rent a dune buggy to cruise around. The Civil Defense Caves and the Ice Caves are very interesting and unique.

Garth’s guidebook
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Shopping for necessities

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Walmart Supercenter
Go back to the Freeway and Head North. Take the next exit and you are at Walmart. There are other businesses as you head south on this exit.
Sightseeing

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WY-22, Jackson, WY, USA
Jackson whole has some of the World's greatest skiing for winter time activities. These include the Tramway and Jackson skiing. Jackson has shopping, museums, Theatres, The Grand Teton Park, and the Elk Refuge. In the summer time, there are the 6 pm shoot outs, Bar J Wranglers entertainment, the myriad of hikes up and around the Tetons. There are boat rides Elk Island for a Trout Breakfast. There are other lake sailing rides and tours. There are several places to go camping in the Tetons.

Yellowstone National Park
Yellowstone is one of the most scenic places on earth with numerous waterfalls, geysers, unique animals including Bison, swans, Pelicans, Elk, Deer, Moose, etc. The Park has hot springs swimming in the Fire Hole Gorge. The scenic boat rides at Yellowstone Lake. Last but not least don't miss out on seeing The Grand Canyon of the Yellowstone.

BYU-Idaho
This University has a great Library for research and study. There are all kinds of University activities and of course there are summer time classes for hobbyists .

Johnny Sack's Cabin
The Johnny Sack Cabin is a cabin that was built by an early settler. The cabin has a water wheel generator that provided electricity for the cabin. Of course it is furnished with furniture that was originally made by Johnny Sack himself. It is built near one of the largest natural springs in the world and is also the start of the Henry's Fork of the Snake River.

St Anthony Sand Dunes ATV Rentals LLC
This is a very unique place to ride or walk through the famous Sand Dunes. You can ride dune buggies, make sand castles, or just walk through the dunes looking at the scenery. It is also close to the scenic caves which are designated as the Civil Defense Caves.

Colter Bay Marina & Boat Launch
Take rides at Colter Bay. Ride to Elk Island for a trout Break fast. Ride to the other side of Jenny lake and walk up the mountain.

Teton Village Trail Rides
Ride the Tramway. Ride a horse through the Teton Canyons. See across the continent.

Quake Lake Picnic Area
See the lake that was created by the Great Yellowstone Earth Quake

Lewis & Clark Caverns State Park
These caverns were discovered by a lonely traveler who fell through the ceiling and was stranded in them for about a week. The sights in the caverns are of a unique beauty.

Salmon River, Idaho, USA
The city of Salmon is the gateway to the White Clouds wilderness Area where there are places that are almost undiscovered by mankind. There is the Salmon river that has some of the best float trips in the world and that includes fishing on the way for Salmon, Steel head and of course Trout. There are the Sheep Eater Indians that weren't discovered until 1959. There are great hunting areas and there are the trails of the Nez Pearce Indians and Chief Joseph.

Swimming and Camping

Green Canyon Hot Springs
Green Canyon Hot Springs has therapeutic hot pools of various temperatures and sizes. There is also camping facilities for summer time use.

Heise Hot Springs Inc
Heise Hot Springs is a place with camping, Golfing, great food, river float trips, and great therapeutic hot springs swimming. In the summer time, there are horseback riding and river float trips. It is close to Kelly Canyon Skiing resort for winter snow skiing. There is a scenic drive along the Snake River with camping as well.

Eagle Park Campground
This is a good riverside road that flows along side the Teton River. It has some camping sites, fishing sites and it connects to the ponds and pathways there.
Entertainment

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Bar J Chuckwagon Suppers
Great food - Chuck Wagon Style Chicken, stake, etc.

Opera House
Sun Valley, Good Food, Ice skating like the Olympics, lots of scenery.

Mad River Boat Trips
Take a great ride down the White water of the Snake River.

Henrys Lake State Park
If you would like to catch a big trout, then this is the place. Fishing in this lake you are likely to catch a trout that weighs at least 5 pounds if not more. You will need to bring a boat or rent one. The Mountain scenery is beautiful as well.