Grover was born to Thomas Grover and Polly Spaulding in Whitehall, New York. At age 12, he worked as a cabin boy on the Erie Canal, where he would eventually become a captain. He married his first wife Caroline Whiting in 1828 and became a Methodist preacher.
Grover moved his family from Kirtland to Far West, Missouri. After Governor Lilburn Boggs issued an order for Mormons to be exterminated from Missouri, Grover and his family fled to Nauvoo, Illinois. Along the way, Caroline became ill. She died on October 17, 1840. That same month, Grover and Caroline's infant daughter Emma died. At age 33, Grover became a widower with five young children to care for. While in Nauvoo, Grover farmed for a living. He was also a bodyguard to Joseph Smith and a captain in the Nauvoo Legion.
In 1841, Grover was called to be a member of the presiding high council of the church in a revelation that now appears in the Doctrine and Covenants. From 1840 to 1844, Grover served three missions for the church in New York, Michigan, and Upper Canada.
Grover was a captain in Brigham Young's vanguard company of 1847 which lead the way for thousands of Mormon pioneers emigrating west. At the Platte River, Grover constructed and managed a ferry that would be used by thousands of emigrants.