...ask 20 different folks how to clean a firearm, and you'll likely get 20 different answers...
Since you asked.... this is simply my method, and is not intended to be the ultimate cleaning regimen, or the best method.
For my Remington 870.....
1. field strip the shotgun... barrel off, and bolt out
2. with Q tips, and patches (lay the patch against the edge of the open bottle, tilt the bottle, and dampen the patch) wipe down the bolt, and insides of the receiver
3. use the brush (like a toothbrush) in your pic to scrub the bolt face until it's clean
4. Drill a hole in the head of an empty shell big enough to accept your cleaning rod.
5. slide drilled case on rod, thread the brass brush on rod end... dampen a patch ( I use a damp patch to apply solvent to the bore. I would not suggest dipping the brass brush into the solvent bottle, as it will contaminate the solvent.) and cover the leading end of the brass brush, and start it into the breech end of the barrel. As soon as the brass brush goes in far enough to seat the drilled shell in the chamber, push the shell into the chamber. ( this will center the rod, and prevent the metal rod from rubbing on the chamber or bore)
6. push the rod through the barrel and take the brush/patch off and withdraw the rod. This will remove any loose fouling. Repeat 5 & 6 until the bore is shiney when looked into with a light source at the other end. Do not try to reverse the brass brush while it's inside the bore, as the bristles are bent a bit and may scratch/gouge the bore if you force them to bend the other way while inside the bore.
There are folks that say a brass brush is not hard enough to damage a steel bore, but I don't use brass brushes unless the bore won't come clean with a patch and solvent. Not saying anyone else is wrong, that's just me. The biggest issue with the bore is not to rub against it with the cleaning rod. In a rifled barrel, be careful of the crown..... (the last portion of the bore the bullet touches on it's way down range... smoothbore shotguns do not have a crown) more rifled barrels have been damaged by improper cleaning techniques than anything else.
Once all the patches come out clean, just wipe the shotgun down with a lightly oiled rag, attach the bore mop (big fuzzy plug) apply 6 or 8 drops of oil, run it back and forth through the bore a few times and you're done. I buy lots of the bore mops, and use some to apply solvent to the bore, and some to apply a finish coat of oil to the bore.
Definition....... "light coat of oil" = enough to detect the presence of the oil, but NOT enough to run. Too much oil in the bore, can run into the action if the shotgun is stored barrel up. This will attract dust, and tend to gum up the action over time. If you shoot a slug out of a heavily oiled bore, you may damage the bore, and possibly the shooter. I use a spray oil, and moisten a shop rag, and wipe the firearms down with it. That particular rag does nothing else but wipe down guns. The spray oil is re-applied as needed. When you're finished, check to see if you can moisten a patch by wiping it on the surface of your shotgun. If you can, there's too much oil on the gun.
Over time you will develop a cleaning regimen, that works for you, and that's great. Cleaning of firearms is a highly subjective subject, and there's no perfect solution for everyone.
Before you clean, you have to shoot, and that's a good thing...