It will look like a dim band of glowing clouds stretching all the way across the sky. It will be low in the west in the early evening but rise higher in the sky later at night. Andromeda will not rise above the western horizon until almost dawn, so you wont be able to see the galaxy. When Andromeda is up, it is possible to see the galaxy as a dim little glowing cloud without any optical aid,......
Good luck. Enjoy the sky!
Edit: As GeoffG said, being up near 49 degrees north with little to no light pollution in late June is a mixed blessing. No lights but the Sun can mess things, taking the edge off the best seeing. On June 21st, you have a 6 minute window from 12:31MST to 12:37MST when the sun will not be a factor. Figure a half hour either side of those times for really good viewing. 10 days before or after the 21st will give you about an hour of the best, darkest viewing. Plan on being out from "dark" (11:00 or so) to around 1:30am.
If you move your vacation to December though, bring a parka and sunglasses because it'll be bitter cold and the stars will be bright. ;^)
You can see the Milky Way from anywhere dark. It will look like a dim band of glowing clouds stretching all the way across the sky. It will be low in the west in the early evening but rise higher in the sky later at night. Andromeda will not rise above the western horizon until almost dawn, so you won't be able to see the galaxy. When Andromeda is up, it is possible to see the galaxy as a dim little glowing cloud without any optical aid, but it needs to be really dark and you have to know exactly where to look.
As long as you can get well away from Kalispell you should normally be able to see the Milky Way. It will appear like an irregular milky band across the sky -- you won't be able to make out individual stars. Similarly, the Andromeda Galaxy looks like a small milky patch.
The bad news, however, is that around June 21 the sky never gets totally dark in northern Montana, because of twilight coming in from the north (midnight Sun). You'll have to stay up till local midnight (around 1 a.m. MST) and then the sky will only be fully dark for a few minutes.
