Educated at University College, London, he was called to the bar in 1849.
In 1803 Palgrave was articled to a firm of solicitors, but was called to the bar at the Middle Temple in 1827.
He was educated at Harrow and at Trinity College, Cambridge, and was called to the bar at Lincoln's Inn in 1786.
In 1721 he entered Merton College, Oxford, as a gentleman commoner, and studied philosophy, mathematics, French, Italian and music. He afterwards studied law at the Inner Temple, but was never called to the bar.
A friend of the family, Lord Foley, provided the funds for his legal training, and he became a member of Lincoln's Inn on his departure from Oxford, being called to the bar in 1730.