If you've got a double, take it from someone who's got 20K+ rounds downrange in one.
You want what's called a Tico tool. It's the long fuzzy thing. If you shoot a lot of shoot clean ammo and don't mind buying a new tico every six months you can simply shoot, then open the gun. Scrub both bbls with a DRY tico and close the gun. Wipe off exterior and put it away. Every few months you'll want to do as indicated below. Or if you shoot 'promotional' ammunition, same thing.
To clean, pop off the forearm, open the gun and remove the bbl ***'y.
Wipe your action assembly down with Break Free CLP (good stuff) and set it aside where it won't get solvent / crud / kicked etc...
Now, take the bbl set and clean the outside somewhat with that breakfree.
Now for the barrels themselves. Take a hook / loop (the kind you put a patch into) rod attatchment, and soak a fairly large (3"x3" for 12ga) patch in solvent. Run it down both bores, twice, taking the patch out and turning it each time. Let the bores sit for a minute or so when you're done.
Now use your brush and a totally clean patch. Force it through gently, twice each side, turning the patch (I'm stingy). Get a new patch and do it 2X more. Run one more new patch through both bbls once.
Now take your tico tool and run it through both bbls until you're happy that they're nice N shiny. You really don't need any oil at this point unless you're putting the gun up for more than 3-5 weeks. If you are, put some breakfree on a patch (like five drops) and run it through the bbls a few times.