Clean Before You Start
Before whipping out your sandpaper and paintbrush, give the piece a thorough cleaning. After years of neglect, it's likely to have a patina of dirt and grime. Removing the buildup can reveal a nice finish underneath, so a good cleaning and buffing may be all that old table needs to look new again.
Give the furniture a thorough scrub with a sponge and some vegetable-based oil soap and warm water. For detailed pieces, use a soft toothbrush to get into the nooks and crannies. To get between tight spots, pick up a slender wooden dowel rod and sharpen it like you would a pencil. Then use the sharp end for digging out grime. If you need to use steel wool on some stubborn wax buildup, go with 0000 grade (the finest grade available), and use a light hand. Too much pressure can dull the finish.