If $A$ is a real symmetric matrix, can I prove that all of the eigenvalues of $A$ are real and that all eigenvectors associated with distinct eigenvalues are orthogonal? If so, where do I start to prove that?
Yes, this is a famous result known as the "spectral theorem". It's a little easier to prove over the complex numbers, but the real symmetric case isn't that much harder. See http://en.wikipedia.org/wiki/Spectral_theorem#Hermitian_maps_and_Hermitian_matrices .