Here are some indications.
Fact 1: If $S_n/n^{1/p}$ converges to a finite limit, then $X_n/n^{1/p}$ converges to zero.
Fact 2: $\mathrm E(|X|^p)$ is finite if and only if $\sum\limits_n\mathrm P(|X|\ge n^{1/p})=\sum\limits_n\mathrm P(|X_n|\ge n^{1/p})$ converges.
Fact 3: For any sequence $(Y_n)$ of independent random variables, $Y_n\to0$ almost surely if and only if $\sum\limits_n\mathrm P(|Y_n|\ge y)$ converges, for every positive $y$.
For the proof of the problem, use fact 1, then the direct implication in fact 2, and finally the reverse implication in fact 3. This gives a stronger version of the result where one does not assume that $\mathrm E(X)=0$ nor that $S_n/n^{1/p}$ converges almost surely to $0$ but only that it converges almost surely to a finite limit.
Hint for the proof of fact 1: $X_n=S_n-S_{n-1}$.
Hint for the proof of fact 2: Start by recalling or reproving the classical equivalence that $\mathrm E(|X|)$ is finite if and only if $\sum\limits_n\mathrm P(|X|\ge n)$ converges.
Hint for the proof of fact 3: Borel-Cantelli lemma with independence for the direct implication, Borel-Cantelli lemma without independence for the reverse implication.