There is nothing wrong with learning on a keyboard. The child can learn the layout of the keys, how to read music, and can have a lot of fun while making some progress. Eventually however, if he/she becomes serious, then a real piano--and a good quality one--is going to be necessary for further progress.
A real piano is quite a different beast from a keyboard or even a digital piano. Fifteen years ago, I bought a very beautiful Kawai upright (1975 BL-31 Made in Japan--not China or Indonesia)--gorgeous mint condition, and a very deep, dark sound with singing treble and an interesting mix of harmonics in the bass. $2000! Previously I had played on a very good Yamaha digital piano, but, after buying the Kawai, I felt a much deeper connection to the music, and my teacher noted a rapid improvement in tone and interpretation.
Kawai and Yamaha are both excellent brands, but if you buy used, you should have a qualified technician--not just a tuner--inspect it. Problems with the pinblock, soundboard and/or action can be very costly to fix later on.
I would recommend getting a 48"--i.e., full sized upright. As Keith Jarret points out in The Piano Book, unless your ceilings are four feet high, lack of space is no excuse for not getting a full-sized upright. Avoid smaller uprights and especially spinets, which many tuners won't even touch.
If you want to try to keep your old piano, get a recommendation for a good technician--not just a tuner--in your area to inspect it and give you an honest evaluation of how much it will cost to bring it into tune, and brief you on any other problems it may have.
I greatly prefer the darker sound and treble sustain of Kawai, but admit that Yamahas are very good pianos. Yamahas typically have a much brighter sound that some people prefer. I have extraordinarily sensitive hearing, especially in the upper registers, and the Yamaha sound wears thin very quickly with me. (BTW, Kawai made the first piano in Japan, and later taught Yamaha how to build them! Yamaha also build outboard motors, motorcycles, and the like!)
If money is no object and your kid becomes serious, get a grand. The action is fundamentally different, allows for faster repition of notes, a greater tonal palette, and offers a true sostenuto pedal--which very few uprights do. The older Steinways (my piano teacher's Model B from 1929 is the most beautiful instrument I have ever heard live), Bechsteins, Mason-Hamlin, and higher end Kawais are certainly worth a look.