Palmer was born in Buffalo, New York. Innis participated as a choir boy at his local church until he joined the army. He entered the United States Military Academy in 1842. He graduated in the class of 1846 and was appointed a brevet second lieutenant in the 2nd U.S. Mounted Rifles. He entered the Mexican–American War as a first lieutenant and was cited for bravery and noteworthy service at the Battles of Contreras and Churubusco. His valor displayed at the Battle of Chapultepec earned for him a brevet promotion to the rank of captain. In March 1855 Palmer was promoted to the full rank of captain in the 2nd U.S. Cavalry.