Harrell made his recital debut in New York in 1971, and a year later played at a Chamber Music Society of Lincoln Center concert. In a review of that concert, Harold C. Schonberg of "The New York Times" declared that "it would be hard to overpraise the beautiful playing" of Harrell, adding "this young man has everything". For the rest of his life, he continued to perform internationally as a recitalist, chamber musician, and soloist with orchestras. Also in 1971, he began his teaching career at the University of Cincinnati – College-Conservatory of Music. He went on to teach at the Royal Academy of Music in London, the Aspen Music Festival, the Cleveland Institute of Music, and the Juilliard School. He served as the Music Director of the Los Angeles Philharmonic Institute from 1988 to 1992. From 1986 to 1993, he held the post of "Gregor Piatigorsky Endowed Chair in Violoncello" at the USC Thornton School of Music in Los Angeles; he was only the second person to hold the title, following Piatigorsky himself. He was on the faculty of the Shepherd School of Music at Rice University from 2002 to 2009.