Which things should one keep in mind when thinking about buying an acoustic piano? Age, wood, brand? What things are high quality or bad quality? What to watch out for?
The most important aspect of buying a piano is to get one that you enjoy playing and that has a sound you find pleasurable. There is no correct answer here and really boils down to personal preference with regard to tone and action. Do you prefer a bright, sharp high end, or a deeply resonant bass? Do you prefer a heavy action that requires great muscle power to play loudly or a lighter action?
Another thing to keep in mind is that the tone of new pianos changes with use and becomes a bit more mellow over time. That said, unless you're planning to buy an extremely high-end instrument, a new piano is likely to be preferable to an old piano (that has been sitting neglected in someone's living room for decades, for example)--recent advances in piano manufacturing technology have made pianos better than ever.
Third thing is that the better the brand, the more the piano retains its value as it ages. For this reason and because presumably you'll become a better pianist as a result of owning an instrument, I would suggest buying the best brand you can afford. Piano manufacturing is incredibly competitive and brands take great pride in the consistency and quality of their offerings--that said, a new Steinway is always going to be more fun to play than a new Boston, which will be more fun to play than a new Baldwin, which (usually) will be more fun to play than a new Yamaha or Kawai. Avoid no-name or lesser known brands if possible as instruments by these manufacturers these tend to have a poor tone and go out of tune quickly.
If you have to buy a piano and you cannot judge the quality of the piano by yourself, you should go with somebody who knows better to try it. This somebody could be a good piano technician or a professional pianist.
Other than that, don't trust cheap brands. They use cheap materials and careless manufacturing which, in the best of cases, will translate in manteinance problems. Pianos are complex, fine machinery that need to be very well made.