My favourite finish is Penetrol Wood Oil.
While the piece is still on the lathe, I complete the sanding down to whatever grit I am happy with, depending on the type of timber. The last grit I use is usually 400, but you must use every grit between what you start with, and what you finish with, as each successive grit removes the scratches left by the preceding grit size.
A good tip when sanding is to make the last strokes with every grit in the direction of the grain, even if the piece is still on your lathe. This helps to get rid of those circular scratches you will not see until you apply the first coat of (any) finish.
Once I am happy with the sanded finish, I wipe down the piece to remove all traces of dust. The first coat of oil is applied freely all over that part of the piece not covered by the mounting method. I continue to apply the oil freely until it seems that the piece has "had enough". While the surface is still wet, or not gone sticky, I wipe off all excess oil as well as possible. The piece should feel nearly dry, but tacky. It is important to make sure there are no nooks and crannies that hold oil that will come out overnight and cause a run. I do not turn the lathe on during this step.
I leave the piece overnight, or for at least 8 hours. With 800 grit wet and dry abrasive, using oil not water, the second "coat" is applied by rubbing the piece all over with the abrasive and oil. I use the oil freely, keeping the abrasive sliding easily over the surface. This creates a fine slurry of oil and wood dust which seems to effectively fill any tiny pores and cracks which are very hard to see until the piece is finished. I am happy to stop rubbing back with the 800 grit when the whole of the wet surface feels silky when I run my fingertips over it. Any "rough" patches can be re-rubbed while the surface is still wet. I don't let the surface go sticky during this operation, but just keep adding oil to the abrasive. When I am happy that there are no more "rough" patches, I simply dry the piece again in the same way as the first step. I do not turn the lathe on during this step.
I leave it overnight again. Before applying the third coat, I simply rely on the feel of the piece to check that no dust has settled during the night to form sharp points on the surface. If I find any lumps, I simply remove them carefully using my fingernail, as the surface is not yet rock hard, and they can usually be removed by this method without marking the finish. If not, I re-sand with 800 wet and dry and oil and let dry overnight. This coat is then applied using a pad made of a piece of lint free cloth like flannelette or singlet material about 60mm square with half a cotton ball wrapped inside. I place a small glass bowl on a tray that sits on the bed of my lathe, pour in the required amount of wood oil, and soak the pad in the oil. It helps to make sure as little air as possible remains in the pad, and simply apply the pad of oil evenly all over the surface of the piece. I squeeze the oil from the pad, and gently wipe the excess oil from the surface using the pad, in the direction of the grain before the oil becomes sticky. At this stage, the reflected light helps me to find any spots that I have missed. The lathe is only turned by hand in this operation.
Again, I leave the piece overnight. The fourth coat and any subsequent coat is applied as the third coat.
Some timbers benefit from application of a grain filler before the first coat, while some timbers need only three coats, and others do need a fourth. The degree of gloss seems to increase with each coat applied after the third. At no time do I run the lathe while applying this finish.
This finish, as you can see, is not for people who do not like finishing, but I find it fairly restful, and good therapy, as well as enjoying a beautiful finish which is long lasting, and seemingly impervious to many household stains and liquids.