Well I'm no expert, but since you're not getting many responses, I'll try to give you a quick rundown of how I clean a gun, based on how I learned from my Dad, and from tips I've seen here on the forum. Your kit should also have instructions.
I've never tried it, but there is an all in one product, which many folks seem to like, called CLP...... as in Clean, Lube, Protect....I think "Break Free" may be a brand name of one. I personally use seperate liquid solvent and lube, Hoppe's brand, just because that's what I started out with as a child. There are also some foaming spray bore cleaners which would probably work well.
To start, if you can disassemble your gun and clean from the breech instead of the muzzle, that would be preferred. Get some liquid solvent into the bore by using a solvent soaked patch, and or, dip a wire brush into the solvent. Run the brush through a few times with the solvent to get out the lead fouling. Someone correct me if I'm wrong, but I don't think it's the end of the world if you don't have the exact brush. Use one that doesn't flop around in the bore, yet not overly tight. NOTE: Use safety glasses when the brush comes thru the barrel, or wrap it with a rag .... because dirty solvent and possibly wire brush fragments can fly.
Next start running patches through until they come out clean, checking barrel occasionally to make sure it is coming shiny clean. I start out with a couple with solvent on them, and then start using clean dry ones, as the barrel starts looking clean. A pointed jag, or the slotted attachment can be used to keep the rod from slipping past the patch inside the bore.
After it's completely clean, use a mop (white fluffy thing), or a clean & oiled patch, to put a light coating of oil into the bore. I use Hoppe's 9 gun oil, and flush all moving parts while working them & flushing oil through them, and then wipe the whole gun down afterward with a clean oiled rag. I try to keep an oily white flannel rag laying around to wipe any gun off after I handle it. Always re oil a gun if it get's wet, or if someone handles it with possibly sweaty hands.
Opinions differ on how often to clean. Some do it every time they shoot, some never, some in between. I think guns are fairly forgiving, and will withstand alot, unless you expose them to water, etc. I only shoot occasionally, so I clean every time. I don't feel you would do any harm if you only clean occasionally, but I personally feel that precision mechanical devices, such as guns, perform more reliably if properly cleaned and lubed, and personally, I take pride in keeping my machines clean & maintained, whether it's guns, Harley's, musical instruments, etc. I guess it's all in how important your gun is to you, and if your life depends on it or not.
For cleaning rods, many like the one piece carbon fiber ones, Tipton seems to be a popular brand. I don't have one yet, so I use a 3 piece brass one. I've heard that aluminum rods are not recommended because they can leave shavings behind.
A bore guide is recommended, to help keep the cleaning rod from rubbing the barrel, but care can be exercised if you don't have a guide.
I have a very old hardwood one piece cleaning rod that I got with a very old S&W. The 97 yr old previous owner was a wood worker, and he made it himself, from a chunk of oak. It works well for handguns and will not scratch them. I use it regularly to push patches through the handguns.
Hope this helps....hopefully someone else will add anything I may have left out.
Josh P
It won't suplant my main tackle box kit, but it can go to the woods.