Richard received his early education at Adair's school at Fermoy, and, after serving an apprenticeship to a surgeon in Ireland, came to London to pursue the more scientific part of his professional studies at the Aldersgate School of Medicine, under the supervision of his brother Jones. He afterwards went to Paris, where he attended the lectures of James Richard Bennett, a private lecturer on anatomy and an Irish friend of his father. In 1828, when Bennett was appointed a demonstrator of anatomy in the newly constituted school of the university of London (now University College) Quain assisted his patron in the duties of his new office.
Richard began his education at Adair's school in Fermoy and then worked under a surgeon in Ireland before moving to London. There, he studied medicine scientifically at the Aldersgate School of Medicine, guided by his brother Jones. Later, he traveled to Paris where he attended anatomy lectures given by James Richard Bennett, an Irish friend of his father who was also a private lecturer on anatomy. In 1828, when Bennett became a demonstrator of anatomy in the newly established University College London, Quain supported him in his new role.