When it comes time to apply the finish, do not work from the can. Stir well, then pour the finish through a disposable paint strainer into a clean glass container. Bringing these surfaces back to their former beauty is a great winter project. Most old varnished tables were coated with a very strong, shiny lacquer. Lacquer is beautiful but becomes soft and gummy over the years.
Dirt gets into scratches, and they become very obvious.
You may be able to repair the surface by just washing it down and deep cleaning it with lacquer thinner. The thinner often softens the finish enough, so that it fills the scratches and looks like new. If you are not satisfied, wet sand the entire surface. You'll need a gallon of water and a stack of 220 or 280 black sandpaper. Wrap the sandpaper around a rubber or wood sanding block, dip it in the water and start sanding.