The above advice about abrasives is spot on for stem (get a set of micro mesh pads from Amazon-with elbow grease and going from medium to very fine you can polish nicely to almost machine-buffed finish).
Oxidized vulcanite can taste nasty! For interior I slide a bristled cleaner soaked in alcohol trough stem, clamp one end to bench or vises,pull the anther end taught and work the heck out of stem back and forth under judicious tension multiple times. Doing the same after with a regular cleaner soaked in good old white toothpaste is good, then a heavy water rinse. Mineral or other oil soak on stem for final, then tshirt polish
For the stummel, I ream chamber, maybe sandpaper on a dowel. Shank: clean as much as possible out of mortise gently mechanically, then alcohol/q-tips, folded cleaners, etc). Draft hole gets brush, the bristles then regular cleaners.
Then I do salt/alcohol treatment on bowl (double over or extra fluffy cleaner in shank, then salt in bowl, then alcohol (I like everclear or other very high proof clear liquor).
Polish briar after cleaning w/wax or whatever you feel best with.