Price was born in Blackwood, Wales. Born with deformed legs, she was operated on age four, and suffered with pain in her legs throughout the rest of her life. Her younger brother, born with a mental handicap, was often looked after by Margaret. The family had ties in Cardigan and north Pembrokeshire, and would spend their summer holiday in Moylegrove and Cilgerran.
Her father, a talented amateur pianist, was opposed to a musical career, and hence she never attended a young Eisteddfod, and was aiming for a career as a biology teacher. Educated at Pontllanfraith Secondary School, near Caerphilly, aged 15, her school music teacher organised an audition with Charles Kennedy Scott, who convinced her to study with him at Trinity College of Music in London and obtained a scholarship for her. Over the next few years, Price was trained as a mezzo soprano.
After graduation, she joined the Ambrosian Singers, performing with them on the soundtrack of the 1961 Charlton Heston film El Cid.
Unrecognised through the normal channel of competitions, instead her now converted father became her champion, writing to opera houses to arrange auditions. Resultantly, Price made her operatic debut in 1962, singing Cherubino in Mozart's The Marriage of Figaro at the Welsh National Opera.
After her father wrote to the Royal Opera House in Covent Garden, in 1962 she auditioned and was turned down twice by musical director Georg Solti who said that she “lacked charm.” However, she was accepted as an understudy thanks to casting director Joan Ingpen, and the forming of a close personal and professional relationship with composer James Lockhart. Solti added a rider to her contract, stating that she should never expect to sing lead in the main house, so resultantly she sang minor roles as a mezzo. But on Whit Monday 1963, her breakthrough came when Teresa Berganza cancelled a performance, and Price got the chance to take over as her nominated understudy - again in the role of Cherubino, a performance that made her famous overnight.
After this, Lockhart convinced Price to take further singing lessons to improve her technique and develop the luminous high scale that made her one of the most popular lyric sopranos of the 1970s and 1980s.