Regarding rifles...
Always clean breech to muzzle. Never put anything down the muzzle except a bare rod. You want to pull dirt and debris out of the rifle, not push it into the breech. For the M1A, read Chapter 5 of FM 23-8.
Use a Dewey 1-piece brass rod at home and an Otis kit in the field. The BoreSnake is excellent. Put a wet patch around the bristles.
Jags are the worst thing ever invented. Every gunsmith has probably had to remove a stuck patch from a bore, because their client was using a jag. There is a reason that the military doesn't issue them.