Oh, man, you got it made. Plenty of time to see a lot of the west. Beartooth is usually open the first week of June, but reliably by the middle of the month. Spectacular by July. First week of June will see tall snowbanks Either side of the road where they plowed through to open it. By July, it’s all open and beautiful up there. Lamar Valley too, which is what you go through getting to the Beartooth from the west. Glacier Park, Going to the Sun Road is awesome too, not usually completely open until late June or July. The Big Hole Valley in SW Montana is one of my favorite places on earth. Amazing. Hyalite Reservoir and Fairy lake near Bozeman are nice too.
If Superior interests you, go further north of Missoula, run 200 up the Clark Fork river valley through Thompson Falls and Noxon. Follow it to Sandpoint, ID, then head north to Bonners Ferry and cut back across 2 through Libby to Kalispell. Beautiful country. A side trip to Yaak would be neat, too. From Kalispell, you can head south through the Swan Valley, several little lakes you can drive up to along there, Van lake is a really nice one. You can cut across through McDonald Pass to Helena, then south to Townsend, and over to Judith Gap. Might check out Lewistown, it’s a nice little town at the foot of the Big Snowies. East and a bit north of there, you will find the Peck Reservoir and the Missouri Breaks. All of it beautiful in different ways. What I laid out after Kalispell sets you up for the northern route back home, too. Cut across North Dakota to Minnesota, or cut south from the Peck Reservoir to Billings and catch 90.
Other suggestions on Wyoming are spot on. Grand Teton is a beautiful place, should try for a sunrise at the Snake River overlook there. Spectacular, but gotta be there before sunup, the colors are quick to change as the sun rises. Hope you have a good camera. Highways 14 and 16 through the Bighorns are awesome. As are museums at Cody. Saratoga in the south part of the state is neat, hot spring pool open to the public 24/7/365, good base from which to explore the Medicine Bow area. Beautiful and easy access. Going out, cut off 70 onto 40 north for Steamboat Springs, pretty drive, and it takes you west to Irish Canyon, or farther to the Flaming Gorge. Skip Salt Lake City, traffic is horrendous there, instead take 191 up to Rock Springs, then 90 west to 30 west. Take that to Montpelier, ID and catch 89 north through the Star Valley, then to Alpine, and into Jackson Hole from there.
Should be a lot of county fairs and rodeos in season while you are out there, you oughta hit a couple of those. In fact, the big rodeo at Cheyenne might fit your schedule. They call it Frontier Days, tickets sell out early. You want to sit on the J stand if you go. Aluminum bleachers cook ya on the other side if it’s sunny.
