Is it worth saving?
With any piece of furniture, the practicality of refinishing eventually comes down to one question: is it worth saving? Once you've found a piece you like, and decided what it is, look at it again to see what kind of shape it's in. Most old furniture is fairly sturdy, or it wouldn't have survived; but chances are it's also taken a beating over the years. Are the legs even? Is the piece sturdy? Does it wobble? Do doors and drawers work properly? Are the joints well made, and have they separated?
Assess the amount of work you'll have to do to restore the piece. Is hardware complete and tight? Are hinges adequate? Are drawer guides or dust panels missing? Is the wood covered with many coats of paint? If the piece of furniture is in fairly good condition, or if it's definitely an antique, it will be worth your time and effort to refinish. If the wood is broken or badly damaged, there are parts missing, or the joinery is inferior, don't waste your time unless the piece is an antique.
How bad does the damage have to be before it makes refinishing impractical? This depends on how much work you're willing to do, but there are a few guidelines for decision-making.
First, look for dry rot or insect damage. Dry rot cannot be repaired; the rotted component must be replaced, and this is a custom job. Insect damage, if the entire piece of wood is not affected, can sometimes be repaired; if this is the problem, restoration may be worth the effort. To check for dry rot and insect damage, push an ice pick or a knife blade into the wood. If there's little or no resistance, the wood is damaged.
Broken parts are sometimes repairable, but not always. If a part is split or wobbly, it can probably be repaired quickly. If it's broken off flush at the joint, the job is more difficult, because a replacement part must usually be custom-made to match the rest of the piece. This can involve expensive equipment or a professional woodworker, and the piece of furniture may not be worth the cost or the effort.
On veneered pieces, the condition of the veneer is very important. Has the veneer separated from the base, or is it damaged? Are there big pieces missing? Separated veneer is easy to reglue if it's intact, but replacing damaged or missing veneer can be expensive. If a large section of veneer must be replaced, the cost may be prohibitive.
If the piece is structurally sound, don't be discouraged by repairable problems. Wobbly joints can be re-glued; missing hardware can be replaced. Coats and coats of old paint, lacquer, or shellac may be concealing beautiful wood -- walnut, cherry, oak, birch, maple. If you like the piece, if it's worth saving, and especially if it's an antique, refinishing is worth all the time and patience you'll put into it.
So do your homework and learn what is and isn't an antique, what are the basics of style, and finally, whether your antique is worth even saving before you invest the time, money and energy to get the job done.