He was the first of twelve children and was educated at various schools in New England, such as Chester Academy of Vermont for four years. He studied law at the Law School of Northampton, Massachusetts receiving his degree in 1823, and then began his own practice in Townshend, Vermont. He married Almira Cadwell in 1827 and they had four children, yet two would die during infancy. He was elected to the state Vermont House of Representatives, but after seeing his siblings move to Michigan Territory as well as receiving advice from former Vermonter and Michigan Territorial delegate, Lucius Lyon, decided instead moved his family there in 1834.
Ransom was appointed by Governor Stevens T. Mason as an associate justice of the state Supreme Court in 1837 and served as chief justice from 1843 to 1848. In one notable issue, he issued a declaration in 1840 that prevented the removal of the Catholic Potawatomi from their lands in southwestern Michigan.
There were two notable immigrations to the state during his administration. A group of Hollanders came to western Michigan, led by Rev. Van Raalte, of the Dutch Reformed Church. They founded the city of Holland, Michigan and later established Hope College. James Jesse Strang led the other immigration, consisting of a faction of Mormon followers. They settled on Beaver Island in northern Lake Michigan. Strang founded a kingdom there with a capitol, St. James named for himself. Strang was even elected to the state legislature twice, but anti-Mormon sentiment and his totalitarian rule of the island led to his assassination.
Because of Ransom's strong anti-slavery position, the state Democratic Party did not re-nominate him for Governor in 1850. He was elected again to the state legislature in 1853 and 1854.
He was also the first president of the Michigan Agricultural Society, which was instrumental in the creation of both the Michigan State Fair and Michigan State University. He served as regent for the University of Michigan, 1850 to 1852 and was a co-founder of the village of Augusta, Michigan.
He died at the age of 61 in Fort Scott and is interred at Mountain Home Cemetery in Kalamazoo.