A question before I give an answer.
Is the wood one that already has an oily content? i.e. rosewood or olive? If the answer is yes then an oil finish is recommended.
If not, then here is a step by step
Sand down to 400. Finish with each grit by sanding horizontally (with the grain) with the lathe stopped.
Seal with a 50% mix of cellulose sealer and cellulose thinners. Apply two coats, After the last coat of sealer and before any further finishing product is applied, rub over the moving condiment with a white synthetic finishing pad to remove 'nibs' of sealer. The Webrax pad is made from aluminium oxide or silicone carbide grains, which adhere to a no-woven mix of nylon fibres. It has a high grit level.
Next, All of my non-oily woods receive several coats of acrylic gloss lacquer. Spraying can be carried out whilst the condiment is still on the lathe but usually because of time constraints I spray away from the lathe in a dust free and well ventilated area of the workshop.
Acrylic lacquer will, subject to the ambient temperature, be dry in 20 minutes. It is recommended that the mill parts are put back on the lathe when dry to receive a de-nibbing between coats. The end- result will be well worth the extra time. Usually a maximum of three coats will suffice.
I like to leave the sprayed items, overnight before beginning the buffing process. I find that this 'hardening' time gives me a better end result than trying to buff immediately.
All of my condiments receive the buffing process. I have three 6" loose stitched mops, which I mount individually on a right-handed threaded polishing mop adapter held in chuck jaws. An alternative is to purchase a Beall polishing system, which comes with all you need to buff any wooden item you may turn.
The first mop will be lightly loaded with Tripoli, which is a gentle abrasive compound that will remove minor scratches. I should stress that this is not a substitute for a rushed sanding phase!
The second mop will contain White Diamond, which is a micro-abrasive polishing compound for removing traces of the Tripoli and to create a deep shine. Do not press the condiment too hard or 'smearing' on the polished surface will occur.
The third and final mop is for applying carnauba wax. Apply a small amount and buff the condiment to create a final polish.
You should end up with a mill, that looks go, feels good and stay that way for a long time.
Am I talking about the wife?????? No a pepper mill !