coach (v.) Look up coach at Dictionary.com
1610s, "to convey in a coach," from coach (n.). Meaning "to prepare (someone) for an exam" is from 1849. Related: Coached; coaching.
coach (n.) Look up coach at Dictionary.com
1550s, "large kind of carriage," from Middle 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 in most European languages (Spanish and Portuguese coche, Italian cocchino, Dutch koets). Applied to railway cars 1866, American English. Sense of "economy or tourist class" is from 1949.

Meaning "instructor/trainer" is c. 1830 Oxford University slang for a tutor who "carries" a student through an exam; athletic sense is 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."