In 1954, he was commissioned in the United States Navy and served for three years, rising to the rank of lieutenant.
Before completing his naval service in 1957, he earned his Master of Arts degree from the University of Pennsylvania.
Moving on to Cambridge University, Baugh earned his Doctor of Philosophy degree at the University of Cambridge in 1961 with his thesis, completed under the supervision of John Ehrman, on British naval administration in the War of 1739–48.
Baugh received a Social Science Research Council Grant in 1966–1967, a National Endowment for the Humanities fellowship in 1977–1978 and a fellowship at the Shelby Cullom Davis Center, Princeton University, 1982–1983.
In 2011, the National Maritime Museum, Greenwich awarded him its Caird Medal in recognition of his distinguished career of scholarship on eighteenth century British naval history.