First step is to sit down with the gun and the owner's manual.
Read the manual cover to cover, then using the manual, disassemble the gun per the instructions.
Buy a shotgun cleaning rod kit. These will have an aluminum rod, a loop tip, a bore brush, patches, and often a wool mop.
Buy a bottle of bore solvent and whatever you want to use for a lube.
Use the rod the wipe the bore clean of all factory grease and dirt.
Use a clean cloth to wipe everything else off.
Use a patch or a piece of clean cloth with a few drops of gun lube to wipe everything to apply a THIN coat of gun lube to everything.
Look everything over to check for major burrs, or anything that doesn't seem to be normal.
If you're not sure about something, contact whoever you bought it from.
Reassemble per the owner's manual and check the operation of the gun's action and safety to be sure everything is working properly.
TO CLEAN AFTER SHOOTING:.
Again disassemble the gun per the manual.
Screw the bronze bore brush on the rod, and use an eyedropper or a hobby paint-transfer bulb to wet the brush with a bore solvent.
(Don't dip the brush or patches in the solvent bottle, this ruins it).
Run the brush all the way through the barrel, from chamber to the muzzle. Pull back through, and repeat 15 to 20 times, keeping the brush wet with solvent.
(NEVER, EVER push a brush down ANY gun's barrel part way, then try to pull it back out. This ruins the brush, and can get the brush stuck in the barrel. Always push it ALL the way through and out the end, before pulling it back).
Remove the brush, and as soon as possible, wash it off with hot soapy water to prevent the solvent from eating it up.
Put the loop tip on the rod, and pull a patch half way through the loop.
Wet the patch soaking wet with solvent and push through the bore.
Let the barrel soak while you wipe off everything fouling may have contacted, like the face of the bolt, and the inside of the receiver.
When the rest of the gun is cleaned and wiped off with a slightly oily patch, wet another patch with solvent and run it down the bore.
Let soak another 20 minutes or so, then run 1 or 2 clean, dry patches down the bore.
Hold the barrel up to a good light and examine the bore for any signs of fouling still present.
This will look like dark streaks in the barrel.
If you suspect the bore is still fouled, use the brush and solvent again to scrub the bore another 15 to 20 passes.
Wipe with a dry patch and check again, until the bore is clean.
Put a few drops of lube on a clean patch and run it thought the bore a few passes, then run a clean patch down the bore.
This will put a thin coat of lube inside the barrel to prevent rust.
Reassemble, and you're done.
When cleaning the bore, pay attention to getting the chamber properly cleaned also. Plastic can build up in the chamber and cause problems.
If you have a barrel with a screw-in choke, follow the manual to remove the choke and clean it and the threads inside the barrel.
Failing to clean the choke can cause it to rust in place.