The class equation

Class equation. $G$ decomposes into conjugacy classes. Central elements are singleton classes. For non-central representatives $g_i$:
$$|G|=|Z(G)|+\sum_i[G:C(g_i)].$$

$p$-group. Suppose $|G|=p^n$ with $n\ge1$. Each index $[G:C(g_i)]$ for $g_i$ non-central is a positive power of $p$ (since $C(g_i)\neq G$). So every term in the sum is divisible by $p$.

Conclusion. $|Z(G)|\equiv|G|\equiv0\pmod p$, so $|Z(G)|\ge p$: the center is non-trivial.