He was born in Kenley, Shropshire, to poor parents. He was effectively adopted by a local squire, Joseph Pryce, when Farr and his family moved to Dorrington. In 1826 he took a job as a dresser (surgeon's assistant) in the Salop Infirmary in Shrewsbury and served a nominal apprenticeship to an apothecary. Pryce died in November 1828, and left Farr £500 (equivalent to £43,100 in 2019), which allowed him to study medicine in France and Switzerland. In Paris he heard Pierre Charles Alexandre Louis lecture.
Farr returned to England in 1831 and continued his studies at University College London, qualifying as a licentiate of the Society of Apothecaries in March 1832. He married in 1833 and started a medical practice in Fitzroy Square, London. He became involved in medical journalism and statistics.