For cleaning I like Shooters choice...
http://www.shooters-choice.com/shotgun-and-choke-tube-cleaner
but if I'm in the field ...or at a tournament away from home ...I'll squirt some solvent in the bore and run a bore snake thru it. When bore snakes get dirty ....all you have to do is run some hot water in a stainless sink ...throw some liquid dish cleaning soap in there... thrash the bore snake around a little ..rinse it off ...and hang it up in the garage / or outside to dry.
I think Bore Snakes are a good product.
--------------
I always take the barrels off - to clean them. I use some Q tips and a patch - with some solvent to clean the breech - and the ejectors, etc.
I always re-grease the ears on the receiver that have contact with the barrel ( the point it swivels on - in an O/U )- and where the forend locks in under the barrel at the receiver to barrel connection - especially the big cross bolt lug that holds the barrel....and the lug that locks the forend in. I like Rig Grease. I use break free on the choke tube threads ...and the ejectors.
About every 6 months I take the stock off the receiver ...and clean and "very lightly lube" the firing pins, springs, etc inside the receiver. Some guns get more blowback than others around the firing pins...some used guns have not had their stock off ever...but even on a new gun, out of the box, I take the stock off to make sure all the preservative, etc ..is cleaned up in there.
If I ever shoot in heavy rain or snow...( torrential rain - when you get caught outside in a tournament or hunting is the worst - even if you have a soft case with you - water gets everywhere..)...then I remove the stock to let everything dry out ...and clean and lube it before I put the gun back together.