The first thing you need to know is what type (size) of piano you want. If it's for home study I'd recommend an upright piano because it's not as expensive as the grand piano and it is a higher quality instrument than the spinet (the shortest type of piano). If your budget can afford a grand piano and you've got plenty of space at home, I'd say go for it.
Although people could try to trick you into paying more than the actual instrument is worth or that they don't know all these technical aspects that give an instrument its worth, you should know the basics of what makes a piano good (or bad).
When buying a used piano it's best to take someone with you who knows how to tell a good piano apart from a bad one (ex: a piano technician or a person who knows more than you do about pianos). If you can't afford a piano technician or don't know anybody who can help you, the next best advice is to try and find out for yourself if it's a good piano or a bad piano.
The first step is to look at the inside and outside of the piano, this will let you know if the instrument has been taken care of or not. By searching the serial number (that is usually on the bronze plate or on the back of the piano) you can find out how old the piano is. You don't want an old piano because the materials from which the piano is built may degrade over time (it depends in what conditions the piano was kept).
The next step is to check the strings. When looking inside the piano, look for rust or new strings. If you see rust on the strings then you should move on to another piano because it's just a matter of time until they break. You might ask: how can new strings be a bad thing? Well, new strings can be a sign that the old strings broke because their quality was low. If that's the case then the other strings might break as well.
Ask the owner when was the piano last tuned. If it was tuned a short period time ago and the piano is out of tune, that means that the pin blocks are loose and you should move on to the next piano. Basically, pin blocks are fine threaded screws (that hold the strings) that go through a whole through the cast iron plate (or bronze plate) into a chunk of wood. If the piano hasn't been tuned for a while it will be out of tune.
To know if the piano is out of tune because the instrument hasn't been tuned for a while and not because there are loose pin blocks, just play the chromatic scale and look for uniformity. What I mean by that is although the piano will sound out of tune, you should hear a consistency in the sound and not sounds that are far off from the ones next to it. So if the piano has this consistency in sound then the pin blocks pass the test.
Next, check for the soundboard. The soundboard is a large vertical thin wood plate at the back of the instrument that has the purpose to amplify the sound of the piano. When referring to pianos, there are 2 types of soundboards: solid spruce soundboards and laminated (spruce plywood) soundboards. Laminated soundboards aren't bad, they're decent but they're not as good as the solid spruce soundboards, so aim for solid spruce.
Also you need to check the soundboard for cracks, you do that by inspecting very carefully the back of the piano. Cracks in the soundboard can occur if the soundboard was hit or because of old age, another reason not to buy old pianos. If the soundboard has cracks, you could still play that piano but the crack would make a buzzing sound, which is annoying and can ruin your performance.
The last step is to check for the action in the keys and how they're working. So, if you find keys on a piano that are stuck after you've pressed them or if the hammers that hit the strings are stuck, that's not really a big deal and can be easily fixed and it won't be expensive to do so.
Basically, rusted pin blocks or strings, or cracked soundboards and so on can be fixed but for most of the pianos it isn't worth it because its a little bit expensive and you could get the same or even better quality for a lower price.
These are the most important aspects to look for when buying a used piano.
Web sites where you can search for pianos:www.craigslist.org
Note: never buy a piano without checking it out!