Lesh was born in Berkeley, California, and started out as a violin player. While enrolled at Berkeley High School he switched to trumpet and participated in all of the school's music-related extracurricular activities. Studying the instrument under Bob Hansen, conductor of the symphonic Golden Gate Park Band, he developed a keen interest in avant-garde classical music and free jazz. After attending San Francisco State University for a semester, Lesh was unable to secure a favorable position in the school's band or orchestra and determined that he was not ready to pursue a higher education. Upon dropping out, he successfully auditioned for the renowned Sixth Army Band (then stationed at the Presidio of San Francisco) with the assistance of Hansen but was ultimately determined to be unfit for military service.