The motivation for asking this question is to test how much of the module theory that I have learned so far I can apply it to the simple rings I know in algebra. I have decided to start with the two of the simplest available rings: $\mathbb{Z}$ and $\mathbb{Z_n}$. Later, I'm planning to ask the same questions for other rings.
Categorize all finitely generated free/injective/project/flat modules over $\mathbb{Z}$ and $\mathbb{Z_n}$.
Here is my attempt to classify finitely generated modules over $\mathbb{Z}$:
Free: $M = \oplus_{i=1}^n \mathbb{Z}$.
Because if $M$ over $\mathbb{Z}$ is free then it's a sum of copies of $\mathbb{Z}$ and since it is finitely generated the number of copies must be finite.
Projective: $M = \oplus_{i=1}^n \mathbb{Z}$.
Because $\mathbb{Z}$ is a PID and the sets of f.g. free modules and f.g. projective modules over a PID coincide.
Injective:
$\mathbb{Z}$ is a PID, a module over integers is injective iff it is divisible. Moreover, a product of modules is injective if and only if each component module is injective. Therefore, I guess the problem of classifying all injective modules over the ring of integers is equivalent to the classification of all divisible Abelian groups. Is it right?
Flat: I have no idea.
Here is my attempt to classify finitely generated modules over $\mathbb{Z_n}$:
Free: The answer is similar. In fact, what I said about integers previously is true for any arbitrary ring in general, I guess.
Projective: I assume it has to do something with the Chinese remainder theorem, but I don't know how to go forward beyond that.
Injective: No idea
Free: No idea.
I think you have an OK answer for $\mathbb Z$, except for flatness, which I'll address below. Perhaps the thing holding you back on $\mathbb Z_n$ is that you have not seen much on quasi-Frobenius rings?
Well, the long and short of it is that $\mathbb Z/(n)$ is quasi-Frobenius whenever $(n)$ is a nontrivial ideal.
There is a theorem that says the classes of projective and injective modules over a quasi-Frobenius rings coincide. Furthermore such a ring is Noetherian, so f.g. flat modules are also projective. (This applies to $\mathbb Z$ as well, completing your solution to the first half a little more.)
So for finitely generated modules of $\mathbb Z_n$, flat, projective and injective modules coincide. Classifying one is classifying them all. You could say that they are direct summands of $R^n$ for some $n\in\mathbb N^+$.