Let $G$ be an abelian group of order $p^n$, where $p,n\in \mathbb{N}_{>0}$, $p$ prime. Show that there exist normal sub-groups $G_i \unlhd G$ satisfying $\{1_G\}= G_n \unlhd G_{n-1} \unlhd \dots\unlhd G_1\unlhd G_0:=G$ such that $G_{i-1}/G_i$ are cyclic of order $p$.
I have proven the existence of an element $g\in G$ of order $p$.
Every finite group has a composition series, see this. Now all you need to know is that an abelian group is simple if and only if it is isomorphic to $\mathbb{Z}_q$ for some prime $q$. Obviously $q=p$ by Lagrange's theorem.