If I can't get at the breech end w/a rod, like my M1 Garand's and M1A's, I use a Bore Snake; I use Tipton carbon fiber cleaning rods and bore guides whenever I can, the CF rods are light and cannot harm the bore, the guide keeps everything aligned and makes for less of a mess.
Light Cleaning (no fouling)- Hoppies #9
Deeper Cleaning (fouling) - Butches Bore Shine
I use bronze brushes, USGI ones in .50, .30 & 5.56, when fouling is present, I use a nylon brush if there is no fouling. I brush the bore sparingly, I run a patch 20+ times the amount I use a bore brush; I use Bore Tech brushes and jags.
If I have a frosty bore I use JB Paste, a non-embedding cleaning compound available at Brownells, rubbed into a cotton patch that's then wrapped around a nylon bore brush, the bore brush needs to be a little smaller than the bore, say if the bore is .30 caliber I use a .270 bore brush, and run it through the bore several times to brighten it up and reduce fouling. If you try this run a nylon brush of H#9 and some patches afterward to clean out the JB Paste. This will typically turn dark bores to frosty and frosty bores to bright, if you're into MilSurp arms this technique can be a game changer for an old battle abused bore, but sometimes there's just no hope, corrosive ammo has killed many a good MilSurp arm.
When shooting cast lead bullets in handguns I use the Lewis Lead Remover tool to get the lead out afterwards.
Patches - Almost always Brownells bulk packs or USGI surplus.