A proof of the strong law of large numbers can be more or less complicated depending on your hypotheses. In your case, since you assume that $E[X_i^2]<\infty$ there is a straightforward proof. I am taking this from section 7.4 of the third edition of Probability and Stochastic Processes by Grimmett and Stirzaker.
First, by splitting into positive and negative parts we can assume (without loss of generality) that $X_i\geq 0$.
Second, using the positivity, it suffices to prove that $S_{n^2}/n^2\to\mu$ almost surely; that is, we only need convergence along that subsequence.
Next, Chebyshev's inequality gives
$P(|S_{n^2}/n^2-\mu|>\varepsilon_n)\leq{E[X_i^2]\over n^2\varepsilon_n^2}.$
Choosing $\varepsilon_n\downarrow 0$ so slowly that the right hand side above is summable, Borel-Cantelli finishes the job since then $P(|S_{n^2}/n^2-\mu| \leq \varepsilon_n \mbox{ for all but finitely many }n) = 1.$
In fact, the strong law of large numbers holds under the weaker hypothesis that $E[|X_i|]<\infty$. There are various proofs in the literature, but every student of probability ought to be familiar with Etemadi's tour de force elementary proof. Etemadi uses a clever truncation argument and similar tools to those above, and only needs pairwise independence of the $X_i$'s, not full independence. Some good textbooks like Grimmett and Stirzaker (section 7.5), Billingsley's Probability and Measure (2nd edition), or Durrett's Probability: Theory and Examples (2nd edition) include Etemadi's treatment.
N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrscheinlichkeitstheorie verw. Gebeite 55, 119-122 (1981)