The best PDE introduction that I have found is by Dr. Pavel Grinfeld on youtube. The link is:
These are really really excellent. He starts with ODEs and then moves to PDEs. He covers the introduction to elliptic, hyperbolic,and parabolic PDEs, heat equation, separation of variables, and Fourier series. Unfortunately he does not cover other common topics like Sturm-Liouville problems and Green's functions, etc. But this is probably the best place to start. After that you can find individual videos on Green's functions, etc.