Delgado was born in Calexico, California, to Emilio Delgado and Carmen Rodriguez Delgado on May 8, 1940.
He was raised in his grandparents' house in Mexicali, Mexico, with his extended family, during part of his childhood. As a citizen of the United States, he would walk daily to attend a public school in Calexico. He began working odd jobs as a ten-year-old and at his uncle's bicycle shop at age 12. His family moved to Glendale, California, when he was a teenager. During high school, three years of which he attended Glendale High School, he became president of the Thespian Club; he played the trombone in the jazz band and he was a drum major in the marching band. He played very low notes on the viola in the classical symphony orchestra, he also played acoustic guitar from his early teens throughout his lifetime, playing in folk groups during the 1960s and his 44 years on Sesame Street.
Delgado was "adamantly and morally opposed" to the Vietnam War, but enlisted in the California Army National Guard for six years, serving domestically. A supply corporal, Delgado was deployed to the Watts riots in Los Angeles, in 1965, where he was "astonished to see weekend warriors being issued live ammunition to use against other Americans."
Delgado began acting professionally in 1968, after nine years of "trying to knock doors down in Los Angeles to get in." That year, he received his first Equity job in a summer stock play starring Martha Raye, and later he was cast in the first Mexican-American soap opera, Cancion de la Raza.
Befriending actor Sergei Tschernisch at Los Angeles theatre company Inner City Repertory, Delgado learned of the new theatre program at CalArts, led by Provost Herb Blau. While already a professional actor as of his 1970 enrollment, Delgado praised Blau's methods, suggesting his avant-garde method was "amazing."
As of 1970, he was the artistic director of the new Mexican-American Centre of Creative Arts, which taught Chicano high school and college students from the basement of the Euclid Heights Community Centre in East Los Angeles. Delgado told the Los Angeles Times: "We are 100% positive in our approach, and we are uncompromising in our belief that our kids will come away thinking of themselves as artists. Nothing is going to stop us from attaining our identification in American society."
Delgado had a guest role in an October 1970 episode Storefront Lawyers before being cast as a series regular in Angie's Garage in November 1970. The new children's series focused on serving Mexican-American children. He was billed as a singer-guitarist. He speculated that producers discovered him through the series.