
There are many locations in Wyoming, Idaho and Utah whose photos you have seen many, many times on the internet (Mormon Row, Snake River Overlook, Oxbow Bend, Schwabacher Landing, Delicate Arch, Mesa Arch, The Watchman, etc.). Pretty much all that differentiates them is time of year and weather conditions. That's not a totally bad thing but... if you are like me, you want to capture that special image at that unique location.
I am going to try to point you in the direction of some of those unique locations that I have discovered over the years.
Eastern Idaho

Fall Creek Falls
How to get there: Along Hwy 26 just west of the town of Swan Valley. Right after crossing the bridge over the Snake River, make a left-hand turn. Follow that road until you see a potential right-hand turn. Don't turn but just past that is a one to two car size "parking" area on the left. You have arrived.
What to expect: A gorgeous waterfall along the South Fork of the Snake River.
Best time of the year to visit: Spring, summer and fall.
Other suggestions: Don't fall off the cliff while photographing the falls!

Lava Hot Springs
How to get there: Take I-15 south from Idaho Falls and drive south of Pocatello. Turn left onto Hwy 30 and head east.
What to expect: A bit of a small resort town with lots of red leaves in the Fall
Best time of year to visit: Fall.

Camas Wildlife Refuge
How to get there: From Driggs, take Hwy 33 west to Hwy 20. South on Hwy 20 to believe it or not Hwy 33. Take Hwy 33 west to I-15. Take I-15 north 4-5 miles to the Hamar exit. Drive north on the road that runs parallel to I-15. Make a left at the overpass and follow the signs.
What to expect: Wetlands area with migrating bird populations. Also some occasional wildlife.
Best time of year to visit: Spring as wetlands dry out into the summer.
Other suggestions: Bring plenty of bug repellent.

Craters of the Moon
How to get there: From Driggs, take Hwy 33 west to Hwy 20. South on Hwy 20 to believe it or not Hwy 33. Take Hwy 33 west to Craters of the Moon.
What to expect: Volcanic fields. Moon-like surroundings.
Best time of year to visit: Spring and summer.
Other suggestions: This is a designated "Dark Sky" area so you might want to plan for some night photography.

Mud Lake Wildlife Management Area
How to get there: From Driggs, take Hwy 33 west to Hwy 20. South on Hwy 20 to believe it or not Hwy 33. Take Hwy 33 west to the town of Mud Lake. Look for the signs leading you to the Refuge.
What to expect: Wetlands area with migrating bird populations.
Best time of year to visit: Spring as wetlands dry out into the summer. Not as much as Camas.
Other suggestions: Bring plenty of bug repellent.

St. Anthony Sand Dunes
How to get there: From Driggs, take Hwy 33 west to Hwy 20. North on Hwy 20 to the St. Anthony exit.
What to expect: Fairly impressive sands in Idaho!
Best time of year to visit: Spring, Summer, Fall
Other suggestions: Sand dunes are most dramatic at sunrise or sunset to enhance the contrast.

State Line Road
How to get there: From Hwy 33 at the Driggs stoplight, take a right onto Ski Hill Road. Follow that until you get to State Line Road. Make a right-hand turn.
What to expect: Farmland at the base of the mountains on the left
Best time of year to visit: Fall and Winter
Other suggestions: Explore some of the side roads heading towards the mountains
Southern Utah

How to get there: From Moab, take the road to Canyonlands National Park. Before you get to the National Park, you will see a sign for the turn that takes you to Dead Horse Point State Park.
What to expect: There are a number of locations for photography both to the left and right of the visitor center.
Best time of year to visit: Spring, Summer & Fall

How to get there: Hwy 24 exit off I-70 just west of Green River Utah
What to expect: Incredible scenery
Best time to visit: Spring & Fall... too hot in the Summer!
Other suggestions: Be sure to spend at least one day exploring the San Rafael Swell. Would suggest staying at the Whispering Sands Motel and dining at Duke's Slickrock Cafe... not many other options but those are very good.

Highway 128
While the Moab area has tons to offer a photographer (Arches, Canyonlands, Dead Horse Point), one that provides some unique photo ops is the drive of Highway 128 along the Colorado River.
How to get there: From Moab, head toward Arches NP. Just before you cross the Colorado River make a right-hand turn onto Highway 128 heading north. When you leave the canyon, you will enter into what is referred to as Castle Valley with many interesting rock formations to the east. Continuing north, you will leave Castle Valley and enter another canyon. I don't find that one overly interesting. YMMV!
What to expect: The first part of the journey you will be driving through a canyon alongside the Colorado River. There are a number of pullouts you might use to take some photos. Once you enter Castle Valley, your best photo ops will be to the east.
Best time to visit: Year around.
Other suggestions: You will encounter a number of roads that head east that you might want to explore... road to drive across the LaSal Mountain range, Onion Creek, Fisher Towers, etc.

How to get there: You can start the drive from either Kanab (south) or Escalante (north) Utah
What to expect: Beautiful southern Utah scenery
Best time to visit: Spring & Fall... too hot in the Summer!
Other suggestions: Escalante is a neat little town. If you really want to treat yourself stay at one of the very nice rooms at Loubird Inn.
Western Wyoming

Cascade Canyon
How to get there: Reached from the inner Park Road. Take the ferry from the boat dock at the South Jenny Lake area. Hike up the trail past Hidden Falls and Inspiration Point. Just past Inspiration Point you will start to enter Cascade Canyon.
What to expect: A beautiful steep walled canyon with Cascade Creek running through it. There are many landscapes compositions to be had and you might run into a moose or bear for good measure.
Best time of year to visit: Summer and Fall
Other suggestions: If you are a hardy soul you might want to head all the way to Lake Solitude at the extreme end of the North Fork of Cascade Canyon. Some wonderful compositions to be had on the way out as well as the way back. Be sure to return in time to catch the ferry back otherwise you have another two miles to go!

Blacktail Ponds Overlook
How to get there: Reached from the outer Park Road. Driving north just past Antelope Flats Road look for a sign on the left of the road.
What to expect: Panoramic view of the Teton Range.
Best time of year to visit: All months.
Other suggestions: You can take photos from up on the hill, but you might want to consider hiking down to the valley floor level. This will yield some more intimate foreground interest to your composition.

Granite Canyon
How to get there: Starting from the Town of Jackson, head south to Hoback Junction. At Hoback take a left towards Pinedale on 189/191. Go approximately 11 miles south and until you see a large sign on the left for Granite Creek Recreation Area and Hot Springs, take a left here.
What to expect: There are some nice mountain views along this road. Granite Creek runs through this canyon. There is a nice waterfall that is nice in the Spring.
Best time of year to visit: Spring, Summer & Fall
Other suggestions: Might want to take a dip in the hot spring fed pool.

Hedrick Pond Overlook
How to get there: Reached from the outer Park Road. The turnoff for the overlook is not marked, but there is a sign for Lost Creek Ranch and the dirt road is just a few hundred yards south of the well-marked Snake River Overlook.
Follow these driving directions carefully: There is a gate and a wide spot to park along the road. The hike is very short to the cookout area used by Triangle X Ranch and the marvelous overlook above Hedrick Pond.
What to expect: The best time to come up there is mid-morning, and the Tetons will be lit and the meadows and hills in between will be very nicely lighted
Best time of year to visit: Spring, Summer & Fall

Old Patriarch Tree
How to get there: Reached from the inner Park Road. Make the turn to the String-Leigh Lake. I park in the small pullover just after making that turn to park my car. It isn't visible from the road and thus it takes a short hike through fairly thick sagebrush to reach it. Get your gear ready and hike across the Park road. The tree is roughly 3/4 mile east of the road. There are three dips in the terrain before you get there. In my experience, you only recognize it from photos you have seen AFTER you walk past it and turn around.
What to expect: One of the most wonderful mountain scenes you will ever witness.
Best time of year to visit: You may encounter some snow along with sagebrush in early Spring.
Other suggestions: Morning mist can add a wonderful mood to your photo.

Pacific Creek Road
How to get there: Enter the East entrance to Grand Teton NP and before you reach Oxbow Bend look for a right onto Pacific Creek Road. Continue on this road searching for compositions.
What to expect: Some unique views of the Teton Range with various options for foreground interest.
Best time of year to visit: Spring & Fall

Pilgrim Creek Road
How to get there: Reached from a road between Jackson Lake Lodge and Colter Bay Village. Make a right onto Pilgrim Creek Road (FS Road 30090).
What to expect: Some unique views of the Teton Range with various options for foreground interest.
Best time of year to visit: Spring & Fall

Red Hills
How to get there: Coming from Jackson towards the Park at the roundabout, make a right hand turn onto Gros Ventre Road. Continue past the town of Kelly and look for a sign that points you in that direction.
What to expect: The Red Hills of the Gros Ventre Mountains are an exposed sandstone deposit that are left over from when the region was under a shallow sea, roughly 50 million years ago. The road to the hills is engineered in such a way that immediately after coming around a small bend in the road, a dramatic view of the hills overwhelms you with a sense of natural beauty and wonder. A pullout is conveniently located at that exact location so that you can fully appreciate the view.
Best time of year to visit: Spring, Summer & Fall
Other suggestions: Keep driving after the Lower Slide Lake Overlook as you might find some other compositions of interest.

Signal Mountain Summit
How to get there: From the inner Park Road just south of Signal Mountain Lodge.
What to expect: Great panoramic view of the Teton Range with Jackson Lake in the foreground. You have to follow the path from the parking lot as far as it goes and maybe a bit farther for the best views as tree growth over the years has started to limit things. During wildflower season, you might be able to get some nice macro shots.
Best time of year to visit: Summer & Fall

South Park Wildlife Habitat
How to get there: Take Hwy 189 south from Jackson about seven miles. The entrance to the park is on the right-hand side of the highway.
What to expect: Some beautiful trees and a stream that runs through the preserve along with wildlife sightings.
Best time of year to visit: Spring, Summer & Fall. Closed Dec 1 - April 30

Spread Creek Road
How to get there: Take a right at the next road past Pilgrim Creek Road. Drive just a short distance and turn your view back to the Tetons.
What to expect: Views with the Tetons as a backdrop.
Best time of year to visit: Spring, Summer & Fall

The Other Schwabacher
How to get there: Reached from the outer Park road. Driving north past Antelope Flats Road look for a relatively small sign on the left of the road. The road quickly becomes an "improved" road. At its end is the parking lot for Schwabacher Landing. The Other Schwabacher Landing (my term not official) parking is on the left before you reach the other lot.
What to expect: Walking along the river look for some compositions that include the river in the foreground and the Teton Range in the background.
Best time of year to visit: Spring, Summer & Fall
Other suggestions: If you can't bear being this close to Schwabacher Landing without taking a few snaps, go for it!

West End of Jenny Lake
How to get there: Along the inner Park Road, make the turn labeled "Lupine Meadows". Just after crossing the bridge, make a right hand turn and follow it until you reach a dead end and a parking area.
What to expect: Lake views with the Tetons as a background element
Best time of year to visit: Spring, Summer & Fall