"How do I tell what type of finish is on the table?"

Try an area that is very hidden on the underside of the table. Use a small rag with solvent.

Varnish is dissolved by mineral spirits, lacquer by lacquer thinner and shellac buy alcohol.

Most connercial mass-produced cabinets and tables are done with laquer because it dries fast. This would be the first guess.

Can you post a picture? Some of us flea market and garage sale addicts are always curius what somebody else found.