Good thoughts so far. I will reiterate a few and maybe add a little.
Used piano from a private party is really a crap shoot, probably as bad or worse than a used car. Find a local tuner and get some input similar to having a local mechanic go over a car although most on this forum would probably not have a mechanic look at a car, know your limitations.
Know what you are trying to buy, professional quality music instruments are expensive, but generally play so much better (back to my car anology kind of like the difference between a Mercedes and a Ford). I am of the opinion that many kids would enjoy music lessons much more if they were not crippled by student quality instruments. Upside is that pro instruments will hold their value somewhat, while the student instrument is probably going to be tough to unload without taking a loss or giving it away.
Probably don't have room for a grand/baby grand, but if you are headed that way either Steinway or Yamaha and the discusson ends there as far as I am concerned.
Uprights would go either Yamaha or Kawai. We bought a used Kawai ~25 years ago from a dealer who warranted it for a period. Daughters have countless hours on it as well as my wife, still sounds good to my ear. Has been moved from house to house twice. Sounded awful both times until it adjusted to the new environment and was tuned a couple of times. Due for another tuning soon.
I would not be so quick to dismiss the keyboard. The Yamaha Clavinova has an action so similar to a piano, my SIL who has a degree in Piano Performance and plays professionally can hardly tell the difference between it and the Steinway grand sitting in her music room. If organ is part of the deal the action is not going to matter nearly as much anyway. As others mentioned, much easier to move, stays in tune, flexible playing, etc...
Whatever you do enjoy the music while it lasts. My daughters still play, but rarely at home now. Miss it.