I assume that you are talking about a faithful action of $\mathbb{Z}/n\mathbb{Z}$ on $\mathbb{Z}/p\mathbb{Z}$. First, there are the obvious lifts of the irreducible representations of $\mathbb{Z}/n\mathbb{Z}$ to $G$. For the rest:
Exercise 1: The induction of a non-trivial character of $\mathbb{Z}/p\mathbb{Z}$ to $G$ is irreducible.
Exercise 2: When are two such inductions isomorphic?
Exercise 3: Now count the sums of squares of the degrees of the characters that you get this way.
I have to say that this is rather tough homework if you were not given any of these hints. But with them, you should be able to do the rest yourself.
Edit: More generally, suppose that $G=A\rtimes H$ where $A$ is abelian. Then all irreducible characters of $G$ are obtained as follows. $H$ acts on the irreducible characters of $A$ by $(h\cdot\chi)(a) = \chi(h^{-1}ah)$. Let $\chi$ be a linear character of $A$. Extend it to $S_\chi=A\rtimes \text{Stab}_H(\chi)$, where $\text{Stab}_H(\chi)\leq H$ is the stabiliser of $\chi$ in $H$ under the above action, by setting $\chi(as) = \chi(a)$ for $a\in A,s\in \text{Stab}_H(\chi)$. Let $\rho$ be an irreducible character of $\text{Stab}_H(\chi)$, lift it to an irreducible character of $S_\chi$. Then $\text{Ind}_{G/S_\chi}(\chi\otimes \rho)$ is an irreducible character of $G$ and they all arise in this way. I will leave it to you to determine when two such inductions are isomorphic, so as not to spoil the homework exercise.
Note that your question is a special case of this, since $H=\mathbb{Z}/n\mathbb{Z}$ acts faithfully on $\mathbb{Z}/p\mathbb{Z}$, and therefore also on its irreducible characters. Thus, $\text{Stab}_H(\chi)$ is trivial in your case, whenever $\chi$ is non-trivial.