Almon played collegiate baseball at Brown University. He was the first overall pick in the 1974 amateur draft, by the Padres. After only 39 games in the minor leagues, he was promoted to San Diego, debuting on September 2 as the Padres' starting shortstop. He made a critical error in the first inning, allowing two runs to score, giving the Atlanta Braves a lead they would not relinquish.
Almon led all major league shortstops in putouts with 303 in 1977 and led the National League in sacrifice hits with 20 that same year. The Padres would trade him to the Montreal Expos after the 1979 season, and he would be signed by the New York Mets midway through the 1980 season. He was picked up by the Chicago White Sox for the 1981 season, which would prove to be his best season. As the starting shortstop for the Chicago White Sox, he hit .301 with 16 stolen bases, ranking 19th in Most Valuable Player voting.
Almon played in the Major Leagues until 1988, seeing playing time for the Oakland Athletics, Pittsburgh Pirates, New York Mets on a second tour of duty, and finished his career with 20 games for the Philadelphia Phillies in 1988.