With a new barrel, most barrel manufacturers will recommend a break in procedure which is a shooting and cleaning procedure to help polish the new barrel to help cut down on powder and copper fouling.
What I like is a one shot and clean for five times then, then 5 shot strings and clean after each 5 shot string for 5 times.
My new barrel break in procedures always follows a Bore Brite treatment.
The way I like to clean a rifle barrel is with a good one piece cleaning rod with a jag that is sized to the caliber for that barrel and use good cotton patches 1 3/8 inch patches from .22 centerfire to 6.5 or .264 caliber and 1 ¾ inch patches from .270 and up.
I use and recommend Butch’s or Bore Tech Eliminator Solvent, these will both do a good job on powder or copper fouling.
I will start out with soaking a patch with solvent, push the wet patch all the way through the bore. Then, take a second patch and soak it then push it through the bore, repeat this 4-5 times then let it set for 10 minutes for the solvent to soften the fouling. Then, run a clean patch through the bore making sure you go all the way through the bore pushing all the grit out the end and not to use a scrubbing action as this can damage the bore.
I repeat this about 3 times to make sure I have pushed most of the gritty fouling out of the barrel.
Then soak the bore again with solvent and let set for 10 minutes. At this time, you can use a brush if you want. I will not use a brush before this as I don’t want to use a brush with any grit in the barrel. At this time I may do a little scrubbing action with a clean patch. You can use either a bronze brush or a nylon brush. Copper solvent will eat up a bronze brush if you are using a strong solvent so when you are done using a bronze brush, rinse it off to make it last longer.
With my own barrels, I do a Bore Brite treatment before I start using the barrel and this will, most of the time; eliminate the need for a bore brush.
With bore cleaning you will get a black color on your patch which is powder fouling and the green color is copper fouling. Sometimes you will get a green color just from the brass jag.
The better the quality of the barrel the less cleaning you will have to do.
Most of your off-the-shelf rifles are hammer forged barrel, these barrels will require a lot more cleaning and you will probably need to use bore brushes to get it clean and sometimes using a carbon solvent might be required.
Custom barrels are either cut or button rifled and are much smoother and will give a much better performance.
After your bore is clean, you will want to make sure the chamber and bolt recess areas are both cleaned. Dewey makes a tool that is designed for these two areas and makes the job easier.
Once everything is clean you want to use a bore conditioner to protect the barrel from corrosion and from fouling buildup. I use and recommend Kroil for a bore conditioner. It’s not only an excellent core conditioner; I also use it for corrosion protection on all blued gun surfaces. I also use it for cleaning stainless steel firearms and their parts.
The final cleaning step is bore conditioner. I put Kroil on a patch and run it back and forth several times to make sure the bore is completely covered and also to make sure the chamber and bolt recess are clean and then I put some Kroil on a 2.5” shotgun patch and with a Dewey Chamber Cleaning Kit to put a coat on the chamber and recess area.