I'm currently going through some OpenGL tutorials and from the tutorials it seems like all movement is handled in shaders. You prep the data in your language of choice, then send that data to shaders, and handle all the transforms on shaders. Am I right in this assumption or am I misunderstanding something?
Yes, you are right in this assumption. All PC GPU's released in last 10 or so years internally works by executing shaders. Most of fixed pipeline is gone. Even if you use fixed pipeline rendering on modern GPU, the driver will converts all that into shaders. Of courser you can avoid all that and use shaders directly.