I generally prefer reading instructions to watching videos. Videos can sometimes show what writing can't, but text is often more efficient and has the advantage of easy skimming.
I am fairly new to guns and I have found that the internet provides no consistent answer to the problem of gun cleaning and maintenance. It basically boils down to removing fouling and burnt powder with a solvent and applying an oil to protect from rust and lubricate moving parts. The stronger more effective solvents tend to damage paint, plastic, and wood and for that reason should be restricted to the bore and raw metal.
Step 1: Shotgun (at the Range)
After shooting a shotgun and while still warm a less involved maintenance method can be applied. This method can be done for both break action and pump shotguns.
Apply CLP (Cleaner/Lubricant/Preservative) to the string end of a bore snake.
Drop the cord through the ejector port and out the muzzle.
Place your foot over the copper end piece to hold it in place and draw the gun up to pull the snake through the barrel. Run the snake through until the barrels are clean.
Clean grime off the breech face and lightly oil the outside of the barrels.
Step 2: Shotgun (thorough Cleaning)
Dip a bore brush in solvent/bore cleaner and run back and forth through the barrel. With a break action this can be done from the ejector to muzzle end. With a pump action you can clean from the muzzle to the ejector ports. Allow the solvent to work in for a few minutes.
Place a patch in a loop and run it through the barrel to remove the solvent and loose material. Run patches until they come out clean. Any kind of lint free cloth will do the job. I've even heard of tampons being used for this step.
Now apply solvent to a brush and clean ejector, hinges, and other metal to metal parts. Do not spray into the firing pins.
Wipe off the solvent with a cloth and get into corners with swabs.
Oil the metal to metal parts that were just cleaned including the hinge pins. As a general rule if it slides grease it and if it rotates oil it.
Remove the chokes. Clean them with solvent, apply oil or grease to the threads, and reinstall.
Apply oil to a swab or patch and run inside the barrel. Run a dry patch through after to remove any excess.
Apply oil to a cloth and wipe a thin layer over the outside of the the barrel and any parts at risk of rust.
Any wooden parts can also be protected with a wipe on wax.
After long use or at the end of the year you can clean out the trigger assembly by spraying or dripping solvent over it. Some people like to use lighter fluid for this step. The assembly can then be blown out with compressed air.
Store the guns in a dry place and never in gun carrying bags/cases.
Step 3: Rifle
Remove the bolt.
Insert a bore guide. This is an optional step, but is a safety measure to make sure the cleaning rod does not upset the rifling.
Place a plastic bottle over the end of the muzzle to prevent splatter.
Put a patch in a loop holder, soak in solvent and run from breach to muzzle. Run out and back.
Dip a copper brush in solvent and run all the way out the barrel and back.
Let the solvent sit.
Brush solvent on the bolt and in the receiver. Wipe off with a cloth.
Place a patch on a jag or in a patch holder/loop and run all way out the barrel to remove solvent and loose material. Continue with new patches until they come out clean.
Look down the bore to make sure it is properly cleaned.
Put oil on a patch and run through the bore. Run a dry patch to remove excess.
Lubricate the bolt and receiver. Reinsert bolt.
Run a lightly oiled cloth over the outside of the gun and barrels.
Store in a dry environment.