Broom began studying the guitar at age 12, taking lessons in the American Folk music style. A year later, he began studies with jazz guitarist Jimmy Carter in Harlem, where he took weekly lessons for the next two years. His interest in jazz began in earnest at age 15 and as a result he began his research, study and practice of the jazz art.
Broom attended the High School of Music and Art (now known as Laguardia High School of Performing Arts), where he played in the jazz ensemble and received an award for Outstanding Jazz Improvisation during his senior year.
Broom began his career while still in high school, performing at New York clubs with Charlie Parker pianists, Al Haig and Walter Bishop Junior. In 1977 he played at Carnegie Hall in a concert with Sonny Rollins and special guest Donald Byrd.
He went to the Berklee School of Music from 1978–79, then returned to New York in order to pursue his career while attending Long Island University. At this time he began working in New York as guitarist for Art Blakey and The Jazz Messengers, Dave Grusin, Hugh Masekela and Tom Browne, eventually landing his own recording contract with GRP Records.