I have trouble visualizing simple harmonic motion, and remembering all the related equations. How should I think about it?

The key is to realize that displacement, velocity and acceleration all vary sinusoidally, with the graph of each being displaced relative to the preceding one by 90 degrees; thinking about these three graphs, and how they describe the motion of a horizontal mass-spring system, will give you an intuitive understanding of the dynamics of simple harmonic motion (SHM).Knowing the defining equation of SHM, a = -w²x, and remembering that x, v and a are all sine or cosine functions, you can derive all the equations by simply applying Newton's laws to the system. Do this once, and see how each equation arises from the underlying physics; this will help you remember them, because you'll know what they mean and why they're true.