Mount Rushmore and Little Bighorn are great stops along the way. I live just outside Glacier National Park, and in years with typical snowfall, the snowplowing crews start plowing the 52-mile "Going to the Sun Road" in April and finish around mid to late June. This weekend we are getting hit with another foot of snow and GNP will see 2-3 feet. This year will be above average snowfall, so I wouldn't expect the entire road opened until closer to the 4th of July.
As far as campgrounds, St. Mary Campground inside the east side of the park is very nice. Camping here will allow you to access the east side of the park even with the road closed at Logan Pass on the continental divide. Camping on the west side, the West Glacier KOA is very nice with plenty of amenities. For a cheaper spot inside the park, Apgar Campground is just inside the west entrance and nicely wooded with great bike trails.
Just know that you cannot pull a trailer over the Going to the Sun Road, you must drive around the southern part of the park on Highway 2. However, you can explore lots of the park in a tow vehicle no matter if the summit is cleared of snow or not.
Enjoy your trip!
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