Often problems that don't seem that big are major problems and vice versa. Keys that don't play are usually not a big problem. Often something has broken or come unglued which is easily fixed. There are a few older pianos with old plastic action parts that are problematic.
Look for notes that sound terribly out of tune when played by themselves. Most of the piano has three strings per note. The strings wrap around a steel tuning pin which is set into a wooden pinblock. When the pinblock goes bad it can't hold the tuning pins tightly and a tuning pin will slip. This leaves one of the three strings very flat to the others. This is not just an out of tune honky-tonk sound, but it will sound like you are playing two distinct notes. A bad pinblock may very well be the end of that piano if it is not a good enough piano to warrant rebuilding. On a high quality piano it may be worth doing a major rebuilding to replace the pinblock as these pianos are considerably more expensive if purchased new and would therefore warrant the work.
Another serious problem is the presence of strange rattles or buzzes. The soundboard, which is the large wooden board you can see from the back of an upright piano or from underneath a grand, has ribs glued on it to strengthen it. Sometimes when the soundboard gets cracks in it the ribs come unglued from it in places. This can allow the soundboard to rattle against the loose rib as it vibrates. This can sound like a speaker distorting when it is played too loudly. Pianos have a wooden bridge which is attached to the soundboard and has the strings running over it. The bridges have two pins for each string to hold the strings in place. Because there are so many pins very close together, sometimes the bridges split and allow the pins to become loose. This allows the strings to rattle against the loose pins. Bridges are often made in sections that can come unglued from each other also causing buzzes and rattles.
Look at the hammers for deep grooves cause by the strings. Layers of felt can be removed to restore the rounded shape to the hammer but eventually there in not enough felt left above the wooden molding to get a good tone. The high treble has the least amount of felt and you can sometimes see that the felt is all the way worn through and that the wood molding is actually striking the strings. Hammer replacement is fairly expensive and the piano needs to be good quality to warrant this work. You can't just replace the felt on the hammers. The felt is put on the moldings in special presses under tons of pressure.
The most important thing is to call a Registered Piano Technician (RPT) to look at a piano before you buy it. You should look at the piano first and be sure it is something you are interested in. “Free or “cheap” pianos can actually be the most expensive if the condition of the instrument will require complete rebuilding in order to be a playable instrument. Ask your RPT to check out the structural condition of the piano. An investment of a service call before buying it can keep you from buying and moving a piano that won't be playable much less an instrument you can be proud to play and own. Find an RPT near you using our searchable online directory.