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.  Shortly thereafter, he enrolled at the College of San Mateo, where he wrote charts for the community college's well-regarded big band and ascended to the first trumpet chair. (A snippet of tape of Lesh on trumpet at CSM can be heard on "Born Cross-Eyed" from the Grateful Dead's 1968 release Anthem of the Sun.) After transferring with sophomore standing to the University of California, Berkeley in 1961, he befriended future Grateful Dead keyboardist Tom Constanten before dropping out again after less than a semester. At the behest of Constanten, he studied under the Italian modernist Luciano Berio in a graduate-level course at Mills College in the spring of 1962; their classmates included Steve Reich and Stanford University cross-registrant John Chowning.  While volunteering for KPFA as a recording engineer during this period, he met bluegrass banjo player Jerry Garcia. Despite seemingly opposite musical interests, they soon formed a friendship. Following a brief period as a Post Office Department employee and keno marker in Las Vegas (initially rooming with Constanten, who soon departed to study under Berio and other members of the Darmstadt School in Europe); a second stint with the Post Office in San Francisco; and a collaboration with the likes of Reich, Jon Gibson and Constanten upon the latter's return from Europe under the auspices of the San Francisco Mime Troupe, Lesh was talked into becoming the bassist for Garcia's new rock group (then known as The Warlocks) in the fall of 1964. This was a peculiar turn of events, as Lesh had never played bass before. According to Lesh, the first song he rehearsed with the band was "I Know You Rider". He joined them for their third or fourth gig (memories vary) and stayed until the end.  Since Lesh had never played bass, it meant that to a great extent he learned "on the job", yet it also meant he had no preconceived attitudes about the instrument's traditional rhythm section role. In his autobiography, he credits Jack Casady (who was playing with Jefferson Airplane) as a confirming influence on the direction his instincts were leading him into. He has said that his playing style was influenced more by Bach counterpoint than by contemporaneous rock and soul bass players--although one can also hear the fluidity and power of a jazz bassist such as Charles Mingus or Jimmy Garrison in Lesh's work, along with stylistic allusions to Casady. Lesh has also cited Jack Bruce of Cream as an influence.

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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.