Some suggestions from a new member:
1. Hoppe's Boresnake. Comes in all calibers, pistol and rifle.
2. Carbon fiber, coated, or aluminum rods should be more gentle on the muzzle than stainless steel. They can pick up grit, so should be wiped off frequently.
3. For really dirty bores, Outers Foulout cleaning system. Can't imagine a .22 needing something this serious.