Gower, aged 13, joined the Royal Navy in 1755 under the patronage of his uncle, Captain John Donkley. He was present at the Battle of Quiberon Bay under Admiral Edward Hawke, 1st Baron Hawke and served under Byron's command on HMS Dolphin from 1764 to 1766. He was promoted to lieutenant, serving with distinction under Commander Philip Carteret from 1766 to 1769.