You say you work better when you have permutations as elements. Well, we can work with permutations. Draw the prism and name its vertices 1 through 6 in some way. Now figure out the reflections and rotations; each one maps the set of vertices onto itself, so you get a permutation, and that permutation uniquely defines the symmetry of the prism. From this point onwards you may forget about the geometry and just work with the permutations.
Now, the generators: I'm not sure what you mean by "the first thing to do is get the generators chosen. I know there are 4..." - a group can be generated by more than one set of generators, and you can choose any set you like I guess if the problem doesn't specify one (you can point out to the prof, TA or author that "the Cayley graph of a group" is not a well defined concept). Now that you have permutations to work with, you may find it easier to pick some generating set and then draw the edges from each group element to its product by each generator.