Firearms cleaning is a subject that tends to get overcomplicated. There are a lot of good products out there and a lot of opinions on the best ways to clean, but I'm going to offer my $.02 on the subject.
You may have heard the adage before from someone in the barrel industry that "they've seen more barrels damaged by cleaning than by shooting..." In my experience, this is probably true. Not from over-cleaning per se, but from improper cleaning. That being the case, how does one properly clean their firearm? Hell if I know, but here's how I do it, and it has worked well for me on precision rifles that have thousands of rounds downrange while still maintaining long-range accuracy. My way isn't the only way, it may not even be the best way, it's just a way that I have been successful with.
Step 1 is safety. Make sure the gun can't go "bang." I can't tell you how many times I've heard or read "I was cleaning it and it went off..." When talking about firearms, that's obviously undesirable. So... magazine out, chamber visually and physically inspected, all ammo away from the firearm and cleaning area. Some other safety considerations are eye protection and gloves depending on what solvents you use and if there are springs that can launch out. I recommend performing cleaning over a tray of some kind to catch anything the pops, rolls, bounces, slips or launches out. This tray will also protect finished surfaces of whatever you might be cleaning on. Firearms solvents will lift the finish off varnished furniture in a hurry - ask me how I know. It can also eat carpet, laminate floors and plastic surfaces. The tray helps contain spills if something gets tipped over too, which happens, so a cleaning tray of some kind is recommended.
Step 2 is using the right equipment. I recommend Dewey coated cleaning rods...
...they're pretty much all I use. They aren't all that expensive but they protect the bore and they are worth it. We use a coated rod as one way to protect the crown and chamber from possible damage by running a hard steel cleaning rod against it, which can scratch or groove it (theoretically) and cause inaccuracy.
Another thing you can do to protect the bore is use a bore guide in rifles. Bolt guns and AR's use a breach bore guide, for guns like Garands, M-14's and some rimfires you need a muzzle bore guide. For pistols, I wouldn't be overly concerned, just do your best to avoid rubbing the rod on the crown, that is, that last portion of the muzzle where the bullet exits the barrel.
In addition to a cleaning rod and bore guide if applicable, you need a chamber brush (nylon or brass/bronze - never steel), and a cleaning jag. A jag has a point on the end for spearing and holding patches or bore pellets and a few grooves in it that help force patches into the rifling to wipe away debris, carbon and copper fouling. Brushes and jags are caliber specific.
You need some cleaning patches. I really like the white flannel cloth patches from Sinclair, but any commercial bore cleaning patch is fine. Just get the right size. Attempting to shove a .30 caliber patch down a .22 caliber bore is a good way to get your cleaning rod good and stuck in the barrel, and can potentially damage the barrel.
Step 3 is using the right solvents and cleaning agents. I am a big fan of Gunzilla. It is environmentally friendly and non-toxic. It doesn't smell like kerosene, in fact, it doesn't smell like much of anything. As people, our perception on things is often based off of our other senses like hearing and smell. Gunzilla doesn't smell strong, therefore you might assume it doesn't clean strong. This is not the case. Gunzilla is a pretty aggressive cleaning agent for getting fouling and carbon out of the bore. It is also considered a CLP or "Cleaner, Lubricant & Preservative," one product that does all three jobs. If you don't like or can't find Gunzilla, there is no shortage of other good cleaning products out there. I would tend to avoid anything with ammonia or that smells like ammonia however. If your cleaning product isn't a "CLP" but just a solvent, you will also need a light oil or lubricant such as Fire clean, slip 2000, Break Free, Tetragun, etc...
Lastly, I would recommend a small tube of some kind of grease like tetragrease.
Once you have your unloaded, checked and double-checked firearm and all of your cleaning equipment and materials I recommend field-stripping your firearm according to manufacturer's recommendations.
On AR's, AK's and Semi-auto handguns I typically begin by using a brush and a little cleaning solution to gently scrub any firing residues, lint and other contamination from the action parts. You don't need to get these soaking wet, just enough to emulsify any contaminants and brush them loose. When this has been done, I wipe the part with a lint-free rag or towel to remove excess liquid. On AR and AK lower receivers, a little compressed air, if available, can help clear debris and excess liquid from the trigger groups. For upper receivers, use a brush and solvent to clean away any carbon buildup, firing residues, etc... On AR's, don't forget to clean the charging handle and charging handle channel. Canned de-dusting agents can also be used for this, just make sure not to tilt or invert the can, which can cause liquid to be sprayed into the action. This liquid can cause frostbite, so it's probably more dangerous to you than to the firearm, but we want to avoid this if possible.
On Revolvers I use a bronze brush with a drop of cleaning solvent or CLP to clean the firing residues off of the front and back of the cylinder, the forcing cone area and anywhere else firing residues or debris are apparent. As before, use a lint free cloth to wipe away any excess cleaning agent.
For Bolt guns and everything else, there's not much ability to reach the action parts without more significant disassembly, which isn't recommended. Clean what you can reach with a brush and cleaning agent, then use a patch or cloth to clean up excess fluid. It isn't a bad idea on bolt guns, particularly those that have gotten wet or get used in humid environments to clean and oil the inside and firing pin if you can get to it. Allowing the firing pin to rust into the bolt can render the rifle inoperative.
After the action is clean, I recommend cleaning the barrel. Barrel cleaning has been over-mystified. It's just not that fancy. I recommend you use a patch and gently remove any firing residue from the crown (muzzle) with a little solvent first, then clean the breach-face and chamber. On bottle-neck cartridge guns, a chamber brush or oversized bore brush helps. You can use a 9mm brush on .223 guns (for example) and a .45 brush on .308's, just don't jam them in though, only use them to clean the chamber. When you get to the shoulder - STOP!
After the chamber is clean, run a wet solvent or CLP patch down the bore, from breach to muzzle if possible. Use a bore guide if you have one and try to avoid letting the rod scrape against the muzzle when the patch or brush emerges. If it happens, it's not the end of the world, just don't let it happen a lot, particularly if using a non-coated or steel cleaning rod. Don't reverse the direction of a patch or brush inside of the bore if you can help it, push it all the way through to the other end, retract the rod, apply another patch and repeat as necessary. After a few passes with a brush... maybe 10 or so, use the jag and patches with a little cleaning agent on them. Again, if possible, push from breach to muzzle until the patches come out mostly clean. How many this will take depends on the type of barrel, how many rounds it has down it, how many rounds it has been since its last cleaning and the type of ammo you use. I would say anywhere from 5 to 20 should do the trick. After the bore is clean, run one or two dry patches down the barrel. If you are using a CLP, that is all you need to do. If you are using a solvent only like Sweets, Hoppe's, or Shooters Choice; you will need to apply a thin coat of oil in the bore. Use some bore oil or CLP, 1 or 2 passes with an oily patch, followed by 1 or 2 passes of a dry patch. The Barrel is now good to go.
Lastly you need to clean the bolt or slide. On AR's, bolt disassembly and cleaning is particularly important, especially in the area of the bolt radius where firing residues can build up into a hard enamel-like deposit. There are a few companies making special tools to clean these areas. I personally use the Magna-matic tool, but others are good as well. Don't get super aggressive with these, you just want to scrape the chunks off, not gouge the part or take off any chrome plating.
On pistol slides, I use a bronze brush to clean the anywhere I see brass left behind, such as the face of the slide around the firing pin hole and under the extractor. I also brush anywhere I can see firing residues. A standard toothbrush will work fine for this, but a bronze brush will remove brass and copper deposits better. These really aren't a huge problem, but clean them off if you don't like how they look. What is most important is to inspect the extractor, to make sure it isn't cracked, missing pieces, jammed up with debris etc... Also inspect the firing pin hole, you don't want any debris in there either. I use a patch with some cleaning agent on it to clean the underside of the slide. The recoil spring and recoil rod, if present, seldom require any kind of cleaning... but a light brushing with some CLP every so often can't hurt.
Now let's talk about lube...
On autopistols, I add a little grease around the barrel and in the slide rails. A drop of oil in the trigger mechanism and in the firing parts in the slide is sufficient to keep them from rusting and running smoothly. Re-assemble the pistol then wipe everything down with a dry, lint-free cloth. Grease and lube on the outside of the gun only serves to hold onto lint and grit. The corrosion resistant finishes on modern defensive firearms don't require the thin coat of oil to prevent rust like blued finishes.
On revolvers, a drop of oil in the middle of the cylinder under the ejector and a drop on the crane pivot.
On bolt guns, I grease the bolt lugs the bolt rails inside the receiver... that's it.
On AR's, I go light on the grease, just a dab along the length of the charging handle on top and the bolt cam. The rest of the bolt, I fairly soak with oil. No oil in the buffer tube and just a thin coating on the components in the lower receiver.
On AK's, just don't even worry about it, a dab of grease on any parts that friction has made shiny will suffice, or alternatively bury it in soft mud.
On commercial semi-autos and lever guns, a little grease or oil on any shiny spots inside the receiver, bolt rails, etc... will do. If they are blued, give those parts a thin film of oil or better, wipe them with a silicone protective cloth, that will prevent rusting and oxidation.
On all guns, after cleaning and lubing, it is important to verify function before after re-assembly. Make sure safety levers (where applicable) work appropriately, make sure the trigger releases the hammer or striker and resets correctly, make sure the followers move freely on magazines and that sights are tight. Aftermarket sights, particularly the ones held in by set screws can loosen. Don't be afraid to check under the grip panels from time to time, particularly on carry guns. Lint from clothing can build up in there into big dust bunnies that can restrict the hammer spring.
So, that's just a few thoughts on cleaning. As I said, it's not the only way, just the way that works for me.