Here's the way I clean and burnish shafts. This method produces a shaft that is incredibly clean, smooth, and slick. Even the nastiest, scuzziest, shafts can be be restored to pristine condition. It is somewhat more labor intensive and requires a lathe or cue spinner.
1.) I clean the shaft, and ferrule using a Magic Eraser dampened (not soaking) in warm water. I immediately dry the shaft with a couple of sheets of paper towel (folded over a few times), using a brisk burnishing motion.
(If the shaft is exceptionally dirty it can be cleaned with Comet or Ajax sprinkled on a damp wascloth, or Soft Scrub, and then wiped down with another damp cloth to remove the residue, then proceed to the M E and drying).
2.) I wipe the shaft down completely with some mineral spirits and dry with a cloth towel or clean rag.
3.) I spin it on the lathe and hit it with some 1000 grit sandpaper, wipe it down and apply a shellac based sanding sealer with a clean dry rag. The sealer dries in no time as the heat from the friction of the cloth. I then hit it with 1500, wipe down to get rid of dust, and hit it again with 2000.
4.) I wipe the shaft down again and apply a liberal coat of cue wax and take a break as the wax dries to a haze. I then spin it up and wipe it down with a small towel until the shaft is nice and glossy and then I repeat the process.
5.) Finally with cue spinning I burnish with leather until the shaft is hot and glossy slick, then I take a piece of folded over paper towel, place about 5 drops of Silk Drops Hair serum on it, and apply it to the shaft and sides of the ferrule. I burnish one more time with a piece of white paper cut from a bag and that's it.
In between I keep a microfiber towel in my case and wipe the cue down often in between turns at the table. I make sure my hands are clean and if they get sticky or anything I use a little hand sanitizer and a small washcloth to dry them. I wipe all my cues down and I remove the excess chalk from the tip before putting them away.