The first recorded instance of US mining (coal,  specifically) was in 1,000 AD. The Hopi Indians, living in what is now Arizona, used coal to bake pottery made from clay. More specifically (for your time period), in 1705, the first copper mine was opened in Connecticut. In 1848, the first recorded commercial US coal production began in in the Manakin area (current day Richmond, VA). In 1844, iron ore was discovered in Michigan and the mining industry in that area has continued to grow to the present time.

The websites listed below have a variety of links that will help you in your search of more specific questions regarding British Colonial Mining in America.