My .02 on new barrel cleaning
I use a 1 piece carbon fiber or coated Dewey rods on all my air rifles and powder burners. Neither rod will damage any steel barrel even cheap junk steel, I really don't think they would hurt a brass barrel as long as the rod was remotely straight with the bore, gota use a little common sense. I do not and will not use any screw together cleaning rod on any barrel I care about and no one should use any rod that is bent in the least bit. It takes next to nothing for a bend in a rod for it to rub while inside of a .177/.17 barrel, a bare steel rod will damage the rifling. I don't know of any new gun/barrel I have got that was spotless including $500 match grade Shilen, Hart, or Douglas barrels. I have misc. tips, mops, jags that I use but all barrels get lapped at least a few patches with JB bore paste. A Parker Hale style jag works the best for keeping the patch stuck on the jag and not loosing it or a nylon brush with the patch over it. A .17 cal nylon brush with a patch over it fits good in a 22 cal. barrel The parker Hale style jags have knurling and are longer vs common jags that have a pin at the end to hold the patch and are smooth that only work to push the patch not pull it back. I have a older Dewey knurled jag set with pointed tips not pin/piercing tips that I have had for 15+ years, best jags ever made.
Break barrels are easy to clean just cock it put the safety on and clean the barrel or disconnect the cocking linkage break the barrel open and clean the barrel, put a piece of tape over the air port hole to keep stuff out of the chamber.. You can put a towel on a table to protect the table and gun and let the barrel hang over the side and clean it, put a bowl or whatever to catch any drips. you can secure the barrel to a table leg if you want just for safety. Use a nylon brush dipped in whatever solvent and run it 5-6 times up and down the barrel. Push a few clean patches through to clean what the brush broke loose, a few wet, a few clean until the patches look fairly clean, use a wet brush again if you need to. Then rub some JB bore paste into a patch, good and covered but not oozing either. Put the patch on a jag/nylon brush poke it in and start "scrubbing" it back and forth in short strokes (3"-4" strokes) and work your way to the muzzle, I do this 3-4 times or a few more. Or push it slowly towards the muzzle and when the drag lessens on the patch it's starting to poke out into the muzzle brake so pull it back until it pulls out the breach about a inch and push it back to the muzzle again, push and pull it back and forth again 6-10 times. Push a few dry patches, a few solvent patches a few more to dry until clean and the barrel should be good other than normal cleaning when needed. JB bore paste is made from plant pulp, using it by hand won't harm a barrel, tooth paste is more aggressive, short of using power tools with it and trying to damage your barrel it's safe.