1550s, "large kind of four-wheeled, covered carriage," from French coche (16c.), from German kotsche, from Hungarian kocsi (szekér) "(carriage) of Kocs," village where it was first made. In Hungary, the thing and the name for it date from 15c., and forms are found since 16c. in most European languages (Spanish and Portuguese coche, Italian cocchino, Dutch koets). Vehicles often were named for the place of their invention or first use (compare berlin, landau, surrey). Applied to railway passenger cars by 1866, American English. Sense of "economy or tourist class" is from 1949.
Meaning "instructor/trainer" is c. 1830 Oxford University slang for a private tutor who "carries" a student through an exam (compare pony in the student slang sense "translation"). Transferred sense in sports, "person employed to train athletes for a contest" is attested from 1861. A more classical word for an athletic trainer was agonistarch, from Greek agonistarkhes "one who trains (someone) to compete in the public games and contests."
All panelled carriages with seats for four persons inside, and an elevated coachman's seat, are designated coaches. The town coach proper, has windows in the doors, and one in each end, the quarters being panelled. [Henry William Herbert ("Frank Forester"), "Hints to Horse-Keepers," New York, 1859]
1825, "the use of a coach as a public conveyance;" 1849 as "special instruction or training for an exam or an athletic contest;" verbal noun from coach (v.).