Boresnakes are good for most purposes. You obviously can't use them to scrub lead, copper, or plastic fouling out of the bore, so I consider them adjuncts rather than ends-unto-themselves.
I personally don't think much of the "kits" out there. They always, in my experience, contain a few useful items and a bunch of junk. Moreover, they rarely have any really high quality items.
In your shoes I would buy at least one good coated rod. Dewey has the best reputation in general. Avoid the jointed aluminum rods, at the least. You can get a .22 caliber rod and accomplish all your cleaning with it, but it really would be best to get one for .22 and another for .30 on up, which should by fine for the shotty gun as well. Get a collection of good quality brushes for your various calibers -- and by good I mean brass bristles and a loop on the end, rather than plastic or steel bristles and the end just chopped off -- and you're most of the way home. A good quality solvent like Butch's Bore Shine or Shooter's Choice is a wise investment, as is a can of brake cleaner, which can be used to blast out crud from the action as well as clean the solvent off your brushes when you're done.
Finally, a good supply of soft cotton cleaning patches in appropriate sizes is important. Avoid cheap shiny patches; they're not good for much. A great technique is to wrap the patches around an undersize or worn out brush. This really gets the patches into the grooves and prevents them from falling off in the bore.
A "standard" technique -- one of many but one that does the job -- is to soak a patch in solvent, wrap it around a brush, and push it through the bore -- from the breach, if possible -- all the way out the muzzle, where it is removed and discarded. Then run the brush back-and-forth about a dozen times, occasionally adding some solvent to it. Now remove the rod, clean the brush with brake cleaner, and wrap it with a clean patch. Push the patch through the bore and again discard it. Repeat with 3 or 4 more patches, until the come out clean and dry. This is generally adequate, with the main exception being a bore that is badly metal fouled. In this situation, the above still applies, but then you run a soaking wet patch through the bore and let it stand for 15 minutes or so, then brush and use dry patches. If after that another solvent-soaked patch comes out green, you repeat the soaking, as many times as necessary to remove the green. Or you get bored, whichever comes first.
HTH!
PS -- it's generally unnecessary to clean the bore of a .22 rimfire. Occasionally blow the scunge out of the action with the brake cleaner, run a dry Boresnake through the bore to sop up any leftovers, and call it good.