There is a large number of distinct (up to module isomorphism) ${\mathbb F}_pG$-modules, even in small dimensions, and I don't think you can hope for any kind of reasonable classification.
I did some quick computer calculations. For $p=2$, the number of isomorphism classes of dimensions 1,2, …, 7 is 1, 5, 11, 28, 53, 111, 199.
For $p=3$ in dimensions 1,2,3,4,5, it is 1,6, 25, 78, 235.
For $p=5$ in dimensions 1,2,3,4, it is 1,8,47,310.
Of course, it is sufficient to classify the indecomposable modules, but ${\mathbb F}_pG$ is of so-called wild representation type, which means more or less that this cannot easily be done!
For example, with $p=5$ in dimension 4, 242 of the 310 modules are indecomposable.
On the other hand, I am by no means an expert in modular representation theory, and it is possible that more can be said.
