What FAL said

I pretty much do everything with a rag and a couple nylon/brass brushes(the toothbrush shaped kind). Cotton swabs where needed.
Depending on the shape, a short cleaning rod and a 12ga brush will clean the inside of some .22 receives nicely.

I don't bother cleaning the bore of .22s, a 12ga sized 'mop' kind of cleaning rod attachment will be good for the bore of the shotgun
I like the multi piece cleaning rods for stuff like this.
I even use them on centerfire rifles but I don't recommend that, get a single piece rod for bore cleaning centerfire rifles.