The first step before you can begin refinishing a dining table is to strip the current finish off. Start by washing the surface with a light dishwashing soap or all-purpose cleaner to remove excess dirt, dust and grime. Next you'll want to apply your stripping varnish; these can be found at any hardware store. If you are refinishing a solid wood table that has been painted, many strippers are applicable for paint removal as well as gloss and stain removal. To remove the previous finish, simply follow the directions on your stain remover. It's as simple as applying your stripping agent, waiting a few minutes and then wiping it off with a rag. For heavier coated wood you might have to apply your solution more than once.
Sanding your surface will ensure that your new stain will adhere to the wood grain. It will also help remove any excess stain that was not removed by the stripping agent. You'll want to use a fine-grain sandpaper and sand with the grain of the wood and not against it. You can also use a belt sander; be sure not to buff away too much of the wood. Because of the chemicals that may likely have been used treating the table the first time, you'll want to wear a face mask so as not to breathe in any fumes.
Applying new stain isn't a difficult process. The trick is to use light coats and be prepared for a lengthy process. Using an angle cut or foam brush, simply apply your stain and then leave several hours for the stain to dry. This isn't going to look entirely professional until you add your gloss coat, which can be applied the same way that you did the stain, with thin coats and plenty of waiting. You want to make sure you have a smooth finish on your table, which will require you to apply a coat, wait for it to dry and then sand down the surface with fine-grain sandpaper.