Moore was born in Glasgow, Scotland, the son of Jean Simson and John Moore, doctor and author. He entered the Navy in 1777 at the age of 13. He was promoted to lieutenant on 8 March 1782 to serve aboard , taking part in the relief of Gibraltar under Lord Howe, and the subsequent battle of Cape Spartel in October. During the peace he travelled through France, but was recalled to serve aboard , , and then , the flagship of Sir Richard Hughes on the North American Station. On 22 November 1790 he was promoted to commander in the sloop , before finally returning to England in 1793.