How many subgroups of order $\displaystyle p^{2}$ does the abelian group $\displaystyle Z_{p^{3}} \oplus Z_{p^{2}} $ have?
A subgroup of an abelian group inherits commutativity, and we know the order of the subgroup is $\displaystyle p^2$, so we can apply the fundamental theorem of finitely generated groups to see that the only groups of order $\displaystyle p^2$ are $\displaystyle \mathbb{Z}_{p^2}$ and $\displaystyle \mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$. All that it amounts to is counting elements of order $\displaystyle p$ and $\displaystyle p^2$ in the two groups, and then I think you can take it from there to count how many subgroups of this order you could make out of these elements.
Hint: Theorem If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is $\displaystyle \phi(d)$ where $\displaystyle \phi$ is Euler's totient function. Euler's totient function - Wikipedia, the free encyclopedia it is the number of positive integers less than and relatively prime to n.