June is too early for Glacier's Going to the Sun road but I'm sure there's plenty else to do. I visited late August 2013 and there was still snow.
In May, the great lakes will be cold. You will likely find snow on the ground in Michigan's UP, and in the deeper woods of the LP.
If you see one thing in Michigan make it the Sleeping Bear National Lakeshore. It's consistently voted as one of the most beautiful places in the USA. Pictured Rocks National Lakeshore in the UP is also quite a sight, not sure about how accessible it is in May.
Other than that Michigan has four National Forests, hundreds of lighthouse and historical Mackinac Island.