He was educated at Repton School, and on 24 May 1780 was admitted as pensioner at Queens' College, Cambridge, where he made the acquaintance of Sir Egerton Brydges, who came up at the same time.
[1][2] About 1785 Shaw went to live at the house of Sir Robert Burdett, 4th Baronet at Ealing, to superintend the education of his grandson Francis.
[1] Shaw was elected fellow of the Society of Antiquaries of London on 5 March 1795, and on 27 April 1799 he succeeded his father in the rectory of Hartshorne.
He made a tour to the west of England in 1788, and published an account of his travels in the following year, having studied the mines in Cornwall.
The book became popular, and was reprinted in John Pinkerton's Voyages and in William Fordyce Mavor's British Tourists (1798 and 1809).
[1] The results of research by Shaw with Brydges appeared in the four volumes of the Topographer for 1789 to 1791 which they edited, and the magazine contained many of his illustrations.