that brass bore brush shouldn't hurt a thing. just get it to a point that it [the patch's] come out as clean as you shoved em in. I never use anything but cool water with just a drop of dish washing soap. then I pump this magic solution thru till I get nuttin but clean white patch's every time. then I run clean dry patch's till no moisture comes out on em. then I spray liberally with Rem oil till the whole thing drips inside n out. lock, stock and barrel ... inside n out.
next I run a clean patch thru the bore to remove most of the oil then wipe down the whole rifle with a nice soft cotton cloth ... the metal and wood alike will love this oil treatment and keep everything fresh till ya want to go shootin again.
have fun with yer new shootin iron and keep it simple yet effective.