I'm writing about how to make a cuboid out of wood... is there a difference when I'm talking about the sides? Can "face" "surface" and "side" all mean the same thing? I know the other parts are edge and corner. If so, is it better to use the same word all the way through the instructions or change the word for variety?
surface is the least appropriate term to use: it really refers to the upper or outside layer of something. You could talk about the entire surface of the cuboid (all six faces) as one surface.
For side, even the major dictionaries disagree about what it means:
flat outer surface of an object, especially one that is not the top, the bottom, the front, or the back Cambridge Dictionary
An upright or sloping surface of a structure or object that is not the top or bottom and generally not the front or back. Oxford dictionary
So, a cuboid could have either two sides (left and right) or four sides (left, right, front and back).
In everyday speech, face is used to describe the front of an object, however in geometry it is used in the way that you require- for example:
A polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. Wikipedia
All of these terms are therefore prone to some confusion, but if you wish to describe each of the six faces using the same term, face is probably the best term to use.