Costa was born in Knoxville, Tennessee, where she lived for much of her childhood. She was raised Baptist and sang Sunday school solos at the age of six. At Knoxville High School (Tennessee), she sang in the chorus. When she was in her early teens, her 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.