Depending on what I'm using for a solvent/cleaner, I will or will not use a brush.
For quicker cleaning (i.e., not overnight), I prefer a bore foam, and brushes are not really part of my regimen with these cleaners.
For Liquid solvents, (Hoppe's original #9, or Butch's Bore Shine), I will use a nylon brush to ensure the solvent application reaches to the full depth of the rifling grooves. I don't use bronze brushes because they deteriorate in the presence of copper removers, and because experience with hard fouling deposits has left me with very little faith in the idea that a brush can mechanically dislodge such deposits in any meaningful way. IMHO, the solvent does the work, and only time can make it work better.
I try to use the largest and most absorbent patch that will reasonably fit into the bore. I figure more solvent in, and more dirty solvent out, the better. I don't use a jag, but prefer a long slotted tip, the longest and skinniest I can find, and stainless or plastic whenever possible. By avoiding bronze/brass/copper in the bore, I can be more confident that color evidence of a copper presence is accurate. By using the long, thin, non-cuprous implement, there is more room far a bigger patch.
Generally, I can manage to clean all my bores with Winchester brand white cotton patches from Wal-Mart, and only use .22 caliber patches for .22 caliber bores, and .30 caliber for all the rest of my bores (.264, .284, .308-311).
Greg