Bore snakes are good for a quick cleaning job. You don't have to worry about damaging the crown like you do with a regular cleaning rod, and they're easy to use. I don't like to use bore snakes, because any deposits of copper or fouling will be absorbed by the snake, and every time you use it you're just putting that stuff back in your barrel. But if you don't shoot much they're just fine. If you shoot a lot, or you're starting to get some copper build-up in the bore, you should incorporate some sort of solvent in your regimen. Birchwood-Casey's 2-in-1 solvent does a good job on copper buildup and powder fouling, but it will also do a fine job of removing the finish on your fine wooden stock

I really like Gunslick's foaming bore cleaner - it seems to loosen copper deposits better, and is supposed to be safe for wood stocks. After you're done cleaning, wipe or spray all your metal parts down with Rem-Oil. It's a very good lube, easy to find, and inexpensive. I keep a 10oz aerosol bottle on my bench and a 4oz aerosol and 1oz dropper bottle in my range bag. If you'll be storing it for very long give it a good coat of Rem-Oil, but for short-term storage just a wipe down with an oiled rag or 'spray and wipe' will suffice. It has teflon in it, so when the oil dries it does a great job of keeping fouling from sticking in your bore, making it easier to clean. It's also safe to use on the stock to keep it looking good, but there are better polishes for wood stocks.
If you shoot very much at all, I recommend a decent cleaning kit. It's best to clean from the breech if possible, but if must clean from the muzzle I suggest getting a bore guide so you don't damage the crown with the rod. I spray down the bore pretty good with the foam, then use a brass cleaning brush to loosen deposits and work the cleaner around, then spray again and wait. After 5-10 minutes use the brush again to loosen deposits and wipe it out with cotton swatches until the bore is dry. If your patches come out blackish-blue, then you've got copper build-up in the barrel. Repeat these steps until you get clean patches coming out. Then run an oiled patch through and you're done.