Harmonics always have frequencies that are multiples of the base frequency. So, if in a result you have a lot of frequencies, you can simply ignore those that are multiples of another frequency which is also present, and strong. Note that after FFT, those frequencies will be approximated, and thus, not exact multiples... But if you have enough sample points, the resolution shoul become fine enough.

Also, a piano has several strings per key, which are tuned in almost the same frequency... which produces the phasing effect you generally hear. The ribbles on strings, especially the lower strings, have a similar effect.

Start experimenting with it. You'll probably get familiar with the results from the FFT, soon enough. Note that if you edit the values from the FFT data, you can convert it back to sound, and hear if your approximation is close enough.