The high country in Wyoming will be cold in early June and the passes may be icy as there will likely be a lot of melting snow from the sides of the roads. I remember a trip we took with our kids to Yellowstone in early June coming up from the south (Jackson). The snow on the sides of the road in the highest passes was higher than the top of our camper which is 10 feet tall. The lowest parts of the park were dry as the snow had melted. Also, the Beartooth Highway is magnificent, but it's not likely to be open until July due to the winter snowpack. The Chief Joseph Highway between Cody and Cooke City is a fun road that will be open. Cody is a fun town to spend a few days. IF you want to see some real desert color country, you might head South to southern Utah and visit Moab, Kanab, Bryce Canyon, Zion National Park, Lake Powell, Monument Valley, etc. The weather there will be warm, maybe too warm at times, but you won't see anything like this area in any part of Europe. You can be there in 2 days from Jackson. If you want to correspond more about this send me a private message as I don't check the forum with any regularity, especially now that we have 14" of snow on the ground where I live and the temp is headed for below zero F tonight.