Palmer graduated from the United States Military Academy in 1936, was commissioned a second lieutenant and served with the 8th Cavalry at Fort Bliss, Texas, 1936–1939.

Palmer served in the Operations Division of the War Department General Staff, 1942–1943; was promoted to temporary lieutenant colonel, February 1943; and was chief of staff of the 6th Infantry Division in Southwest Pacific operations in World War II, 1944–1945.

Palmer was promoted to temporary colonel, January 1945, and permanent captain, June 1946, and major, July 1948; commanded the 63rd Infantry Regiment in the Korean occupation, 1945–1946; was chief of plans and operations of the First United States Army, 1947–1949; was instructor of tactics and then director of instruction at the Infantry School, Fort Benning, 1949–1951; concurrently completed the basic airborne course; and graduated from the Army War College, 1952.

Palmer was secretary of the general staff and chief of the Plans Division, United States Army Europe, 1952–1954; was promoted to permanent lieutenant colonel, July 1953; was commander of the 16th Infantry Regiment, 1954–1955; served on the faculty of the Army War College, 1955–1957; and was deputy secretary of the General Staff and White House liaison officer, Office of the Chief of Staff, 1957–1959.

He was promoted to permanent colonel, June 1961, and temporary major general, May 1962; was chief of staff of the Eighth United States Army, Korea, 1962–1963; was assistant deputy chief of staff for plans and operations, 1963–1964, and deputy chief of staff for military operations, 1964–1965; was promoted to permanent brigadier general, February 1963, and temporary lieutenant general, July 1964.