The above photo shows everything you need. As always, remember there is no magic.
Brass/bronze brushes will not hurt nor wear out the steel metal of your guns – they are gentler than a bullet being fired through the barrel. Brass/bronze is softer than steel. So use a brass/bronze brush whenever you clean your guns. Nylon brushes are nearly worthless for cleaning barrels.
Using cleaning patches/cloths alone will not clean your guns’ barrels. For guns with a rifled barrel, they’ll only clean the lands and not the grooves. You need a brass/bronze brush to clean the grooves. For a shotgun with a smooth bore, brass/bronze brushes are still better compared to nylon brushes.
Let the bore cleaner soak in the barrel for a few minutes before scrubbing with a brush.
If possible (impossible with revolvers and a pain with some long guns), clean and brush from the breech end of the barrel. This is to protect the muzzle end with the crown.
It’s fine to brush back and forth at first because fowling is softer than the barrel, and won’t damage the barrel. Your scrubbing is negligible compared to the wear from shooting bullets. You know the fowling is softer than the barrel because you can shoot bullets and powder through the barrel. But later while cleaning, you should brush in one direction only to push the fowling out of the barrel.
On a particularly neglected gun, upsize the brass brush.
You don’t need a cleaning rod guide. Use your fingers to guide the rod as shown in the photo below.
A basic Outers gun cleaning kit works great. Tighten the sections of cleaning rod hard so they don’t get loose while you clean.
Wrap your screwdrivers and picks in a cleaning cloth so as not to mar the finish of your guns. A flathead screwdriver or pick are handy for cleaning, especially crevices.
After cleaning, oil as much of your gun as you can. Use small amounts of oil. Oil collects dirt, sand, lint, and other debris, which can cause jamming. So wipe or barely drop oil on all bits of metal and any parts that move.
Occasionally disassemble the action and trigger assembly although it’s more work, and then clean and lightly lube everything. Occasionally is okay because the action and trigger assembly get dirty much slower than the barrel.
You can see which areas of your guns need oil by the finish wearing off.
If you let your gun rust, use OOOO steel wool soaked in oil to polish off the rust.
Gun oil that doesn’t gum is ideal, but any oil is better than no oil.