Costa was born in Knoxville, Tennessee, where she lived for much of her childhood. She sang Sunday school solos at the age of six. When she was in her early teens, Costa's family relocated to Los Angeles, California, where she completed high school and won a Music Sorority Award as the outstanding voice among Southern California high school seniors. Following high school, she entered the Los Angeles Conservatory of Music to study with famed maestro Gaston Usigli. Between 1948 and 1951, she appeared with Edgar Bergen and Charlie McCarthy on the Bergen radio show. She also sang with Dean Martin and Jerry Lewis in concerts at UCLA, and made numerous commercials for Lux Radio Theatre.

In 1952, after meeting people at a party with her future husband, director Frank Tashlin, she auditioned for the part of Disney's Princess Aurora. Walt Disney called her personally within hours of the audition to inform her that the part was hers. In 1958, Costa was called upon to substitute for Elisabeth Schwarzkopf at a gala concert in the Hollywood Bowl, conducted by Carmen Dragon. Because of her glowing reviews from that performance, she was invited to sing the lead in her first fully staged operatic production, The Bartered Bride, produced by the renowned German producer, Carl Ebert, for the Los Angeles Guild Opera. Ebert later requested that Mary appear at the Glyndebourne Festival, where she made a debut.