June 16-27 yu have quite a few options.
The weather at that time of year should be nice but if you are at high elevation above 7500 feet you can see some overnight snow....no blizzard..more of a dusting to 2 inches.
Some of these locations would be Yellowstone, Bryce Canyon, and the mountains of Colorado.
The question for you is which part do you want to see?
here are some sampling itinerarys of your travels based out of different airports.....
17-27 is 10 days...
LAS
day 0...fly into Vegas drive to Hoover dam,,,spend night in Williams, AZ
day 1... drive to Grand Canyon
day 2... Grand Canyon...night in Tuba City
day 3... Navajo/Hopi Reservations in AM, Canyon De Chiele in the PM Monument Valley evening sunset. nightthere or in Kayenta, AZ
day 4... scenic drive day up UT 93 through Capitol Reef down US 12..night in Escelante
day 5...explore Escalante area...night at Bryce Canyon
day 6... Bryce canyon for day...night in Kanab/Page
day 7... Page, AZ for the day..do Antelope Canyon and boat trip on water..night in Kanab or Marble Canyon
day 8...Zion National Park
day 9...Zion National Park..drive to Vegas
day 10...
the extra Day could be a day in Sedona, second day eploring Hopi/Navajo area, or another day at the Grand Canyon.
SLC
day 0...fly into SLC drive to Idaho Falls/West Yellowstone
day 1... Yellowstone
day 2... Yellowstone
day 3... Yellowstone AM Grand Teton PM
day 4... Grand Teton AM...drive to Rock Springs/Vernal
day 5... AM drive to Moab..Arches NAtional Park PM
day 6... Arches/Canyonlands
day 7... Arches Canyonlands in AM...drive Monument Valley for PM...night in Cortez, CO
day 8... Mesa Verde National Park...night in Cortez
day 9...Mesa Verde in AM...drive US 550 in PM to Ouray.
day 10...drive back to SLC.
alt day 7... canyonlands/Arches in AM, Evening in Monument Valley, night in Page, AZ
alt day 8... Antelope Canyon and water trip..night in Kanab
alt day 9...Bryce Canyon for the day..night in Bryce Canyon
alt day 10...drive back to SLC via UT 12 to Torrey.
DEN
day 0...drive to Estes Park
day 1... Rocky Mountian National Park..night in Pueblo
day 2...drive through Colorado to Montorse for night..side trip to Sand Dunes National Park and Gunnison NAtional Park
day 3...US 550 AM...to cortez Mesa Verde in PM
day 4...Mesa Verde NAtional Park
day 5... MesaVerde in AM...Moumnet Valley at sunset..night in Moab
day 6...Arches/Canyonlands
day 7...Arches/Canyonlands
day 8...Arches/Canyonlands...drive to Glenwood Springs, CO
day 9...Aspen/Marroon Bells through Independence Pass then drive to Breckonridge...return to Denver
day 10...
day 0...Into Denver drive to Salida, CO or Pueblo, CO
day 1... Sand Dunes National PArk...drive down to Espinada, NM
day 2...Bandalier National Monument AM, PM Santa Fe
day 3...Santa Fe...night in Grants, NM
day 4...Acoma Pueblo in AM drive to Canyon De Cheile PM, sunset in Monument Valley
day 5...Arches/Canyonlands
day 6...Arches/Canyonlands..night in Cortez
day 7...MesaVerde National PArk...cortez
day 8...Mesa Verde AM, PM US 550 through Ouray...night in Glenwood Springs, CO
day 9...Aspen/Marroon Bells...drive to Estes Park
day 10...Rockey Mountain National PArk...to Denver
day 0...to Estes Park
day 1...Rocky Mountian NAtional Park...drive to South Dakota
day 2...Black Hills Area (Badlands National Park, Custer state Park, Mt Rushmore, Wind Cave National PArk)
day 3...Black Hills
day 4...Black hills
day 5...AM Black Hills...drive to Billings, MT or Codey, WY for night
day 6...Yellowstone
day 7...Yellowstone
day 8...Grand Teton...drive to Rock Springs, WY
day 9...drive to Aspen through Rock Springs/Vernal (Flamming Gorge, Dinosaur NAtional Monument)
day 10...Aspen/Marron Bells...drive to DEnver