Buying furniture secondhand is the most reliable way to get something better made than the equivalent new budget allows, and the reason people hesitate is that they are not sure what they are looking at. The checks that matter are physical, quick and require no knowledge of periods or makers.
Lift one end. Weight is the fastest single indicator of construction, because solid timber is heavy, plywood is moderate and particle board is light and oddly dense at the edges. A chest that a person can pick up alone is usually not solid wood. That is not automatically a reason to walk away, but it changes what the piece is worth and what can later be done to it, since veneer over board can be painted but not sanded back.
Then look underneath and behind. The back of a cabinet and the underside of a table are where nothing was ever dressed up for sale, and they will tell you what the piece is made from and what has been done to it. A solid back panel screwed into rebated sides is a good sign. A thin sheet pinned across the back is neutral and normal. Fresh screws, mismatched timber or a patch of new glue means a repair, which is worth knowing about rather than worth avoiding.
Open every drawer all the way and close it. A drawer that runs smoothly on wooden runners and stops evenly has been made and maintained properly. A drawer that drops as it comes out has worn runners, which is a straightforward repair. A drawer whose front is coming away from its sides is telling you the joints have failed, and if one has failed the others are on their way. Look at the joints while the drawer is out: interlocking cut joints indicate a level of construction that a stapled butt joint does not.
Take hold of the piece and rock it gently. A chair or table that flexes at the joints under light pressure will keep flexing and will eventually loosen completely. On chairs this is normal and usually repairable by knocking the joints apart, cleaning and re-gluing, but it is work, and the price should reflect it. On a piece where the frame is glued and pinned rather than jointed, movement is a warning to leave it alone.
For anything upholstered, press down hard on the seat and listen. A frame that creaks is a frame that has a problem. Feel for the edges of the frame through the padding to see whether the timber is where it should be. And be realistic about smell: fabric and foam hold odours and damp indefinitely, and the cost of a full reupholstery on a secondhand chair is frequently more than the chair is worth unless the frame is genuinely good.
Two final checks that are easy to forget. Measure the piece and measure the route into your house, including the turn at the top of the stairs, because furniture from an era of larger rooms is often larger than it appears in a shop. And check for active insect damage in old timber: old exit holes with a grey, dusty interior are historic, while clean sharp holes with fresh pale dust beneath them are current and mean the piece should not come into the house until it has been treated.