My technique:
Field strip. Leave the upper handguard. (They break easily during removal.) Scrub everything down with Ed's Red, including the bore, and let it for a bit. Blast it off with whatever brake or carb cleaner was on sale the last time that I was at the store. Lubricate. (Sparingly.) I grease the bolt roller, bolt lugs, and hammer face if I have grease available; otherwise, oil works fine. After this I go back and finish the bore.
I strip the bolt every once in a blue moon, more to refresh my memory of how to do it than out of actual necessity. I use the GI rod handle/ combination tool to do this because I'm too cheap to purchase a better tool and when I was doing it for a living, Uncle Sam never provided one either. A certain amount of proficiency (and tolerance to pain) is required. It comes with practice. Reassembly is worse than disassembly but neither is impossible. If you've never done this before, keep an eye on that ejector. It will, in fact, eject.
I do not tinker with the gas system. It's self cleaning, self regulating, and largely foolproof unless a fool like me tinkers with it. If the piston moves under the influence of gravity (with the op rod removed, of course) the hex nut stays on. If I believe that oil or bore cleaner got into the gas cylinder, I'll strip it down to dry it; if I have to strip it down, I'll clean the piston out of habit while I have it open. I have never seen a piston that actually needed cleaning in order to function properly. I have seen pistons worn out from overenthusiastic cleaning with wire brushes.
I keep the rifle upside-down while cleaning. If you have bedding or a nice stock this will keep solvents away from it. It also will keep such things out of the gas system. Be sure to bottom the rear sight before setting the rifle on something upside-down. A hard knock on an elevated rear sight can (and always did, when I was on duty) shift the zero at best or strip the teeth at worst. Don't get me started on what can happen to the front sight.
I can't help you with any tricks for dismounting the op rod; the disassembly technique for an actual M14 is somewhat different. With the M1A, if in doubt, jiggle it. This also works nicely for getting the bolt back into the receiver.