What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.

I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$

I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as far as I got.

Maybe, this will help. Let's suppose the samples are taking from a normal distribution. Then using the fact that $\frac{(n-1)S^2}{\sigma^2}$ is a chi squared random variable with $(n-1)$ degrees of freedom, we get
$$\begin{align*}
\text{Var}~\frac{(n-1)S^2}{\sigma^2} & = \text{Var}~\chi^{2}_{n-1} \\
\frac{(n-1)^2}{\sigma^4}\text{Var}~S^2 & = 2(n-1) \\
\text{Var}~S^2 & = \frac{2(n-1)\sigma^4}{(n-1)^2}\\
& = \frac{2\sigma^4}{(n-1)},
\end{align*}$$

where we have used that fact that $\text{Var}~\chi^{2}_{n-1}=2(n-1)$.

Hope this helps.
