How to polish up/restore gleaming surface to chest of wooden drawers?

use vinegar to remove stains (well diluted in tepid water) blot dry then leave overnight to dry right out. buff with a soft cloth if no shine then add only a small amount of a light oil (like a baby oil - unscented) or if you must a tiny amount of furniture polish to the cloth and buff thoroughly, other than that seek help from a Professional or you could ruin a really good piece of furniture