OK -
Out of tune doesn't matter so much as long as its not really terrible, in which case you will hear it, believe me! You will need to get it tuned after it's been moved anyway.
Check every single note on the piano, white and black. Make sure they all work. All of them should bounce back up after you've played them. If there's any you're unsure of, play the same note repeatedly as fast as you can- it should play these repeated notes without sticking.
Even one sticky note can be quite pricey to repair. Walk away. If there are problems on the higher or lower registers, or only the middle has been tuned (astounding, but it does happen!), walk away.
Next play the notes again and make sure that the sound stops once you have let go of the keys. Notes that keep on sounding after you've let go of them can mean serious damper repair work unless it's only one or so at most. Walk away.
Now play a whole host of notes, 10 if you can, all at the same time. Any notes will do. While you are holding them down, depress the right pedal. Then let go of the notes. The sound will continue. Then let go of the pedal. The sound should stop completely. If it doesn't, walk away. Repairing pedals is pricey.
Now depress the left pedal. Sometimes this can cause the keys to shift or depress slightly, which isn't a problem in iteslf as long as they all move evenly. If some move down and some stay sticking up a bit, walk away.
Have a look inside the piano if you can at the soundboard (the wooden board behind the strings). there should be no cracks in this board. This is very important.
Finally check the outer for signs of any liquid spillage. This can do serious damage to a piano.
What I can't teach you is to listen for tone. But if all of the above are OK, you will have a piano that is fine to start with, and by the time it needs replacing your budding pianist will recognise a good tone.
If you are confident enough to buy from the paper you will often get a better deal. Piano shops sometimes sell the most awful rubbish at inflated prices - depending on the shop, of course. But you need to know that buying from a shop is no guarantee that what you get will be any better than from the paper.
Hope this helps!
Once you have it, by the way, let it settle for a month and then get it tuned. Don't put it by a radiator or in a very dry or overheated room. Happy buying. 