Stacy began his theatrical career in Taft, Texas, playing Old King Cole in the local school pageant. He was six years old, but he remembers his costume being made by Mrs. Hess, the lady who lived next door to his grandmother's house. It was a large white furry collar with black birds sewed into the cloth, and, in his own words, said: "I'd put that on, and then donning my crown, I really felt like a King. I guess I just liked being somebody else."

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He began acting in school plays in Junior High and at Van Nuys High School in the 1950's, and majored in Drama and English at UC Berkeley in the 1960's. Stacy spent a summer at Tufts Arena Theatre between his college sophomore and junior years, working at the Oregon Shakespeare Festival, and gained his first National attention from critic Henry Hewes, then writer for The Saturday Review. Hewes called Stacy's portrayal of Henry the Fifth and Mercutio "masterful".

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He won a scholarship to the Yale Drama School in 1964 and that summer made his professional debut, cast as Marcellus and The Player King in Joseph Papp's production of Hamlet in New York's Central Park. After Yale, he was awarded a Fulbright Scholarship to London's Academy of Music and Dramatic Art (1965-66), and on his return to America was accepted into the Lincoln Center Repertory's maiden season at the Vivian Beaumont Theatre.

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Stacy's rendering of Macbird!, a satirical off-broadway dark comedy, written by Barbara Garson, suggesting that Lyndon Johnson (in the guise of Macbeth) killed John F. Kennedy. It won him an Obie for his performance. Joseph Papp approached Keach to play Falstaff in both parts of Shakespeare's Henry IV. Soon afterwards he returned to Lincoln Center to play the Bastard, Edmund, in Lee J. Cobb's interpretation of King Lear.

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The following year, Spring 1969, Stacy went to the Arena Stage in Washington, D.C., to star as Buffalo Bill in Arthur Kopit's Indians. The play was a huge success and came to Broadway later that year, Stacy receiving a Best Actor Tony nomination for his performance.