We discuss collective excitations of a trapped dilute Fermi gas within a hydrodynamic approximation. Analytical results are derived for both high- and low -temperature limits and are applied to 40K and 6Li systems of current experimental interest. We identify spectral signatures which can be used to detect the onset of Fermi degeneracy. Also, we find an interesting class of internal excitations with an unusual spectrum. Some of our results are relevant to the case of trapped bosons as well. Our analysis suggests several experiments which address fundamental problems of collective motion in quantum fluids.