Don't use a plastic bore brush, because it won't do the job well. Plastic brushes are too soft to dig through the fouling inside the barrel. Likewise, don't use hard brushes such as stainless steel, because those are too hard and are likely to damage your gun. Remember the scraper discussion? Same principle.
Given the chance, clean from the breech (rear) end of the barrel. This helps reduce the chance of damaging the gun's crown (if it's rifled) - and it also makes it easier to start the brush, because the rear end of the barrel is almost always larger than the muzzle, even when the chamber is not made integral with the barrel.
Apply some solvent to your gun's bore, or to the cleaning brush. Here is where a spray-type solvent shines, because you can squirt a little into the barrel or onto the brush. Never dip the brush into the solvent. Doing so will pollute your nice clean solvent with all the nasty stuff that your brush has cleaned out of barrels in the past.
Clean That Bore
Run the brush through the gun's bore - all the way. Then pull it back through. Never reverse direction with a metal-bristled brush when it's inside a gun's barrel. Why not? Because the bristles are leaning backwards as you push the brush through the bore, and when you stop the brush and pull it the other way, the bristles have to bend to allow the brush to travel that direction. Once that happens, your brush is just about worthless for its intended caliber, because its diameter is reduced and it just won't clean well at all.
Allow the brush to rotate with the gun's rifling, if rifling is present. Many cleaning rods have handles that swivel for that reason.
Next, use a jag to push a clean dry patch through the bore. After that, I will often turn the patch over and push it through again.
Ideally, you will repeat the brush/patch process until the patches always come out nice and clean. I have actually done that, but only on rare occasions. Most often, the patches will begin to look clean and then I'll give the bore a good dose of solvent and brushing, and they'll be nasty again, so I end up getting most of the fouling out and stopping when I get tired of the process.
It Doesn't Have to be Perfect
The fact is, making a gun's bore perfectly clean is difficult, and is almost always unnecessary anyhow (speaking only of guns that shoot smokeless powder; always thoroughly clean all the fouling from black powder guns, because it's corrosive). So get rid of the worst of the fouling and clean the bore until you're tired of doing so or until it's clean, leave the bore with a light coat of some kind of rust inhibitor inside, and you should be in good shape.
If the gun is a revolver, run your brush through each chamber in the cylinder. You may need to use a slightly larger brush, or wrap a worn-out brush with a patch, to get a good snug fit in the chambers. On other types of guns, be sure to clean the chamber well. This is a very important part of the gun, especially on semi-automatics.
A Word on Patch Jags
Listen - I'm a cheapskate at times, but even I appreciate the value of a good jag when cleaning any gun with rifling. The slotted patch holders that came in most gun cleaning kits are almost worthless. When you're patching out a gun's bore, you want the patch to rub against the bore snugly and uniformly, in order to remove fouling. You just can't accomplish that with one of those el cheapo patch holders.
Get a good caliber-specific jag for each caliber being cleaned and a good supply of cotton cleaning patches, and you'll be able to clean your gun well. And if you prefer, old t-shirts often make good cleaning patches, if you're willing to spend the time cutting them up.