When cleaning any rifle; ALWAYS USE A BORE GUIDE. You can purchase bore guides that fit either end (muzzle or breech) it centers the rod, and protects from nasty scratches that can damage a muzzle or chamber.
The generally accepted procedure (after thoroughly cleaning) is to run an oil soaked patch down the bore, then a dry one. This leaves enough oil to protect, but not so much that it's messy, and running into the action.
I cleaned the bore on my first rifle a few weeks ago (yes, I know that's shamefull) It's a remington 581 I've had since I was a kid. I shoot it once in a while, usually it's a loaner gun for learners. I couldn't hardly get the bore snake down it (dangly lead would not get down the bore!). When the snake came out, had black chunks on it!! I scrubbed the bore (back and forth, repeatedly, with the brush, soaked in Montana Extreme Solvent) finally got the chunks out, ran 1wet 1dry (alternating) with Butch's bore shine, after a pile of patches they started coming out clean.
Then it took 2 clips (14rnds) before the accuracy came back to where it was before!
Bottom line: clean the gun regularly, but expect it to take a few shots before it shoots like it did.