The most important concept to keep in mind when cleaning a shotgun is safety. Always be safe while handling firearms. Always treat the gun as if it were loaded and always keep the muzzle pointed in a safe direction. And always remember to check, maybe twice, that the shotgun is unloading before beginning to clean it.
With safety in mind it is also important to have the proper and adequate materials. For shotguns, general gun cleaning supplies are acceptable. A person will need a few rags, cotton swabs, pipe cleaner, brush, bore snakes (not required but very handy), action cleaner, rim oil, rubber gloves, and a cleaning mat.
With all the proper materials at hand a person needs to know their gun. They need to know how their gun works and how it is dissembled. They can find this information in the manual for their gun or on YouTube.
The first major step in cleaning a shotgun will be to take the barrel of the shotgun off. Next, if the person has a bore snake, they will want to drop the brass weight of the bore snake down the back of the barrel so it comes out of the muzzle, then grab the weight and pull the string tight. They will want to be in an environment where powder and dirt can fall to the ground (family living room is not the best choice) and pull the snake through, then inspect the barrel. They will probably need to pull the snake through a few times to remove most of all the foreign material from the inside of the barrel. Next they take the powder solvent and spray it into the action, getting all places that are dirty, then let the solvent sit for 15 to 30 seconds and then dump remaining liquid out. This liquid will likely be dirty. After that, spray cleaner in again and repeat until the liquid the person dumps out is clear solvent. Now it is time to reassemble and lightly oil all parts of the gun, including the barrel (interior and exterior), action (interior and exterior), and it may be appropriate to spray a little extra oil on the main moving parts of the gun. If the gun will be introduced to colder climates a person will not want to use much oil at all. When the oil gets cold it hardens and works against itself. Lastly, a person wants to wipe down the gun and store it in a safe place.
Students should keep in mind that failure to properly take care of a gun can cause the gun to rust and not function properly in the future. Rust and other undesirable consequences of improper gun care can be prevented, but once they are there it is almost impossible to revert the damage.