I vaguely recall something about a ship he served on between the Farragut and Enterprise, but can't think of the name. It wouldn't have been mentioned on-screen, though, so maybe I'm thinking of something I read in The Making of Star Trek
But I'm guessing that at one time he must surely have captained sailboats on Earth, or at least in his bathtub.
In "Court Martial", it said Ensign Kirk logged Finney's mistake while serving on the USS Republic. The Farragut, when he was a Lieutenant, was mentioned in "Obsession".