Cleaning is like any other gun.
You can use a rod, brush, patches, and bore solvent, or you can get by with a bore snake and solvent.
Truth is, you can't get a gun barrel really clean with a bore snake, but it'll do good enough.
Clean out the bore with whatever rod or snake and a good bore solvent. Most any bore solvent at the gun shop or Walmart will do.
Make sure you get the chambers clean. Many shotgunners buy a special chamber cleaning brush on a short handle to insure the chambers are really clean.
After cleaning, dry the bores then apply a thin coat of lube to protect. After applying the lube, run a clean patch or bore snake to remove the excess lube.
Use cloth or paper towel and solvent to wipe off the breech face to remove any carbon fouling, then apply a light coat of lube to protect.
Lubricate the hinge the barrels pivot on.
Wipe the exterior with a light coat of lube and wipe off the excess.
There isn't a lot of info on the Savage-Stevens-Springfield doubles. They made a LARGE number of models and variations of double guns, too many for most sources to go into detail on.
Take note that many of these guns were budget models made long ago, and even if its in really nice condition, it might not be safe to shoot with modern higher pressure ammunition.
Have a good gunsmith look it over before shooting it.