Pam,
Thanks for the pictures of the bore and cylinder chambers. They actually look pretty good for a gun that old that digested black powder. But, I recommend cleaning them and other places on the gun that look rusty with a cleaning product called Blue Wonder Gun Cleaner. It will remove the rust but leave the finish alone...even on a blue gun.
I recommend removing the cylinder from the center pin before cleaning. That way you can work on the bore from the rear of the barrel and not accidentally damage the muzzle. There is a cylinder catch screw on the top strap just in front of the barrel catch. That should be loosened to lift the cylinder catch up so the cylinder can be unscrewed and pulled backward off the center pin. You can see the cylinder catch which is an L shaped plate under the latch that is screwed down until its lip drops behind the cylinder and forms a stop to prevent if from being pulled off the center pin. As you loosen the screw, the catch will lift up and allow the cylinder to move rearward. Just grasp the cylinder and pull it rearward then unscrew it from the center pin...turn left to loosen, right to tighten it when you put it back.
Blue Wonder has several videos on YouTube that show how to use the cleaner on the bore to remove lead, powder and copper residue. You basically apply it to a bore brush then run the brush up and down the bore 7 times then let sit for 15 minutes. I would do this for all the cylinder chambers as well. It's a lot easier to clean the cylinder when it is out of the frame. Then, push a balled up paper towel through the bore/chambers until all the cleaner and gunk are removed. (When you push the towel through, make sure the paper wad exits over an old cloth or some paper to catch the gunk that comes out.) Once clean, put a light coat of oil on the bore/chambers with an oiled cotton patch and it'll be good to go hold your doors open.
Frankly, I hesitated writing this because we are giving you so much information. It's really a lot easier than it sounds even though you might feel like you're drinking from a fire hose.
