OK, here goes.
One piece cleaning rod(s) with good handle bearings (I prefer Dewey, coated 44")
Cleaning rod guide that has a bushing that stays on the rod and plugs into the back of the guide.
Appropriately sized piercing, and Parker Hale style jags.
A dozen brass core bronze brushes
A few nylon brushes.
Square flannel patches of a size that does not require too much force to push a wet one through a dry dirty barrel
Cleaning solvent, strength depending on the quality of the barrel, with the understanding that less quality equals more jacket fouling, which requires a stronger solvent to remove efficiently. For custom hand lapped barrels, I prefer Butch's Bore Shine.
Some sort of plastic squirt bottle to hold and apply solvent, one that will not deteriorate from solvent, or cause the solvent to change.
Jar of JB Bore Cieaner, and tube of IOSSO
Action and chamber cleaning tools with supplies.
Specialty grease for locking lugs and cocking cam.
Roll of ordinary kitchen paper towels
Plastic box for cleaning supplies.
Cases for rods ( I use two, one for brushing and the other for patching.)
If you have the budget, a bore scope will teach you a lot, if not, a friend who has one will be a big help.
One also needs some sort of cleaning vise, and with that some sort of patch catcher can be handy. I also recommend the use of scope caps to protect the lenses while cleaning.
When I clean, I first make sure that the rifle is unloaded, remove the bolt, and place it in the cleaning vise. Then I insert the rod guide and secure it, an cover the comb of the stock with something to keep the solvent off of it, to protect the finish. Then I put a patch on my jag, and insert it into the back of the guide until it is within the solvent port, and use the solvent bottle to wet it thoroughly. If the barrel is very dirty, with more than about 25 rounds fired since the last cleaning, I slowly push the patch down the bore, just far enough so that it can be removed after it clears the muzzle. Then I repeat, but this time I short stroke the patch vigorously in the throat of the chamber and on down the barrel, being careful to slow down and be more careful as I approach the muzzle. All the time that I have the rod in the bore I take great pains to keep its handle in perfect line with the bore, and not to bow the rod. If the patch is too tight, I remove it and trim it with some scissors that I keep in my cleaning kit. After perhaps running 5-6 patches, I wet a bronze brush (in the port with the bottle), and cycle it up and down the bore perhaps 15 or 20 times, being very careful to not extend the brush farther past the muzzle than I have to to reverse it without jamming the bristles. I follow this with 2-3 very wet patches, just passed through and removed at the muzzle, (otherwise the ground up bits of bronze from the ends of the bristles will give a false reading as to the condition of the bore with regard to jacket fouling). Then I let the wet barrel sit for a few minutes, and if the next wet patch looks good, I dry the bore with patches, use the action kit to clean and dry the chamber, and clean the lug recesses. Then I put a very thin film of bolt grease on the backs of the locking lugs, and a larger amount on the cocking cam. I do this every time that I clean.
Every hundred rounds or so, after my normal cleaning, but before doing the action and chamber work, I saturate a patch with light gun oil (nothing fancy) and work some JB into it before stroking it up and down the barrel, without letting it go past flush with the muzzle except when I am finished, to remove it. I use a Parker Hale jag with the patch wrapped for a snug but not tight fit for this. After 10-15 cycles I use solvent to clean all of the abrasive our of the bore and chamber, and then do my usual chamber and action work. This is not a complete description of all of my cleaning options, which vary by barrel, powder, and caliber. I did not get into the nylon brushes and IOSSO because that would be a topic of its own. I have not edited this, but simply rattled it off of the top of my head, so pardon the rough edges. Barrels do not have to be spotless, just clean enough so that the fouling does not stack up from cleaning to cleaning, back to a uniform starting place. Factory barrels will have to be fouled more shots than smoother custom ones after cleaning to get to a stable condition needed for best accuracy. The number of shots needed will be greater after abrasive use. Be very careful with abrasive use, and with rod technique. Every time that I take a rod completely out of the barrel, I wipe it off, and I do not lay rods on concrete benches, or dirty tables. I keep them in a homemade double rod case between uses. When cleaning at the range, I take pains not to drip solvent on the bench, floor or a table. I have built a large rang, so it irritates me to see people not take care of facilities.
Boyd