I must have messed up either the 3D cursor or the origin point, as every time I try to click and select, to move the object or rescale, the origin point moves along with my mouse wherever I click?
I am pretty sure you're using the Cursor tool (red) instead of the Select Box tool (green). Switch to that one and let me know if everything works.
Note that the 3D cursor and the Origin of an object are different things.
The 3D cursor determines the point in which objects are created in the 3D environment. Read: https://docs.blender.org/manual/en/latest/editors/3dview/3d_cursor.html?highlight=cursor
The Origin of an object is the pivot point where the object rotates. Read: https://docs.blender.org/manual/en/latest/scene_layout/object/origin.html?highlight=origin