As a short cut, here are some quick links for SHM videos:
Basic derivation of sine and cosine solutions
How to construct a solution with phase angle
The simple pendulum and small-angle approximation
A stick oscillating due to spring and small-angle approximation.
For kicks, if you want to see where a quantum number comes from, we can use SHM of an electron and Schrodinger equation . This is kind of cool!