Review what you already know about the furniture. You may have checked the provenance when you bought it --- that is, one or more documents stating who the previous owners were and the origins of the set. Check whether the furniture is reproduction or antique. If it is a reproduction, that does not mean it is not valuable, and it may still be quite old. There was a fashion for reproduction furniture in the Victorian period.
Check the lines of the furniture and the quality of the workmanship. Old mahogany furniture may have inlay, sometimes of another wood, such as rosewood, or it may be decorated with gold leaf. Check the legs and casters of the tables. You may find that the casters are brass, and the legs have been elegantly turned, using a lathe. This usually denotes a better quality and more expensive piece of furniture. The quality of the work may give you useful clues about the value of the set. Much antique furniture is modeled on the Chippendale style, with features such as lattice patterns on chair backs, leaf, acorn or rose patterns and leather or other luxury fabric upholstery.
Study the market. The demand for antique furniture is subject to change and is unpredictable. It is a limited market because large pieces of dark furniture do not suit everybody's house. Attend auctions and visit sales rooms. Study antiques guidebooks and compare your dining room set with other pieces from the same period. Ask a reputable antiques expert to assess the value of your furniture. If you choose an independent assessor, it will cost you a fee.