Cleaning firearms is fairly straightforward. Basically you need a cleaning kit which you'll find at any sporting goods store that sells firearms or ammunition. The kit consists of gun oil, liquid powder solvent, an aluminum cleaning rod, bore brush, and some cloth patches. There will be specific instructions for each kit, but the basics are the same. Also, it probably won't come with a whole lot of cloth. You can use a fine fabric in place of this. Old t-shirts are perfect.
Basically you'll use the solvent with cloth patches and the bore brush to remove residue from the black powder. The gun oil is used sparingly on the action of the firearm to keep all the moving parts well oiled and clean. Finally, a lot of folks also like to use a silicon cloth to wipe down the gun after you're done cleaning. This is mostly aesthetic and removes fingerprints, spots, etc..., which could be desirable for your a staged production, but your LD may not be happy as it will make the firearm shiny
. I was going to get very detailed, but found
this website that sums it up quite well and has some pictures and tips, and is a much better explanation than I could provide.
The primary reason for cleaning a firearm is safety. When shooting a firearm, black powder residue is discharged from each round. This is also true in blanks, the only difference between a blank and a live round is that there is no bullet, only a cartridge of black powder. Over time the residue can build up to a point where it causes an obstruction in the bore. As I'm sure you can imagine, an obstruction in the bore of a gun while shooting is not what one would call desirable. Residue also builds up in the action, and if not cleaned regularly the action will not be as precise as it needs to be and can eventually lead to malfunctions of the weapon, misfires, and jams.
Best practice is to clean your firearm after every session of shooting to remove the residue immediately. Now, thinking along the lines of theatre, I'd say that's probably overkill. You won't have any rounds passing through the bore of the stage gun so you don't need to worry about that issue, and you'd be hard pressed to build up enough black powder residue in the span of a week to disrupt the action. I'd say a weekly cleaning would be more than sufficient for the amount of shooting you'll be doing.
Hope this is helpful!