The kit that is available at the above web address is the place to start. This kit is probably available at one of your local gun dealers. Use cotton, not synthetic patches. Wrap the patch around the bore brush to get the best fit. A tight fit is important to get the bore clean.
I run the bore brush through the barrel form the breach 10 times. Wet a cotton patch, 12 gage in your case, around the bore brush, and run it through the barrel from the breach end. Wet another patch and repeat. Wrap a dry patch around the bore brush and run it through the barrel. If it comes out clean, you are done. If not, repeat the entire operation.
Clean the action as directed by the manufacturer. Lightly lubricate the moving parts with several drops of gun oil. DO NOT USE WD40. Put a little gun oil on a cotton cloth and wipe the exterior metal parts.
I will supply additional information if your shotgun has removable choke tubes.