Before getting into sound quality, one thing that can set pianos apart is their action. Last time I knew a lot about what was going on with pianos, only pianos with horizontal strings (grand style) could have a full proper double escapement action. That affects how quickly you can play the same note again after you've played it at least once, and/or how far you have to release a key before you can play it again. This isn't super important for all pianists, but some pieces can be demanding in this area (e.g., the middle section of Samuel Goldenberg and Schmuyle from Pictures at an Exhibition).
For sound quality, I can think of three broad categories that almost anyone should be able to discern between pianos.
Brightness: This one is probably the easiest to hear and will mostly be related to the hammers and the felt thereon. One important thing about selecting the brightness you want in a piano is that in general, pianos get brighter as they age because the felt on the hammers is compacted. So consider looking for a piano that is slightly less bright than you like.
Low end depth: This one is mostly going to be about how much money you have because getting a real boom out of the low notes pretty much requires longer strings. This is why people play 9 foot grand pianos when they can.
Finally, harmonic richness: This will also be related to your budget but more in the way in that you can buy into a better level of richness and once you get up there the differences become much more subtle. Already this category starts off as the least obvious to the ear. To me I hear the difference in the "middle" of the sound. What is going on besides just hearing the notes and the brightness and low end. This quality will also often mature because it has a firm basis in the sound board and its design and materials. Wood matures through its whole life so as the sound board ages the sound should become richer. You might try playing some pianos at a store and talk to the staff about tone quality and sound board construction.
I recommend playing a few pianos that are out of your price range completely to get a sense of what the more affordable pianos are trying to compromise on. You'll want to be able to choose the compromises you make.
To me there's a break point around five of six feet in grand pianos where at that size and up you have a real serious instrument for life. For some of the smaller baby grands you might compare them to higher end uprights because for less money and a lot less living room/music room space you can get quality sound while only sacrificing action.
If you're on a super tight budget you should compare digital pianos. I don't know if anyone makes spinets any more but I can't see spending money on one instead of a high quality digital.
One more difference you can find in piano construction is the implementation of the quiet pedal, sometimes called "una corda". That means "one string" and the classic design for the pedal is that the whole keyboard and hammer action slide to the side so the hammers only hit one string instead of two (in the main range, two instead of three on the high end and glancing on the one string for the low notes). This is another area where you often need a grand body to put in the authentic action but I have seen at least one upright with authentic una corda action.
Enjoy your search! Definitely get out and play as many instruments as you can. Be the customer! Play around, ask questions, and learn. If it's not a lifetime investment this time around, hopefully it will be the next time.