Michigan is not a cave state, there is only a little limestone and only one single show cave. But Michigan has a long history of iron and copper mining, which continues until today. Beneath those two main resources, gold, silver, gypsum, slate, salt, coal and limestone have also been mined. So there is a wealth of mining museums, show mines and also the possibility to visit a working mine.
Adventure Copper Mine
Bear Cave
Caledonia Copper Mine
Cliffs Shaft Mine Museum
Delaware Copper Mine
Iron Mountain Iron Mine
Kitch-iti-kipi
Mackinac Island State Park
Millie Mine Bat Cave
Quincy Mine Hoist
Republic Iron Mine
Stevens Twin Sinks Preserve
Tilden Mine
Wattson and Wattson Jewelry Store
Search DuckDuckGo for "Mine Cave Michigan"
Attractions in the Iron Mountain MI Area mentions some underground sites.
Mine Headframes and Structures of Michigan
Mining History Links, a very detailed link list of mining history related sights.
Iron Trail Convention & Visitors Bureau, this trail leads to many iron mining sights.