Have I got this correct -
Say we have a population. We take a random sample of size $n$ from this population. I.e. we form a sample $S$ based on random variables $X_1, X_2, ..., X_n$ taken from this population. So the sample $S$ is a set consisting of of these random variables, and hence the mean of the sample $\overline X$ will be a function of these random variables.
The distribution of $\overline X$ will be approximately normal. And if we keep taking samples infinitely, and standardizing $\overline X$ by taking $$h(x) = \frac{\overline X - \mu}{\frac{\sigma}{\sqrt{n}}}$$ the limiting distribution of $h(x)$ will be $N(0, 1)$.
Is my understanding correct here?
Yes, you're essentially correct. There's one important thing you've implied by saying "random" but haven't stated explicitly, which is that the random variables must be independent. (It is often stated that the variables must be identically distributed as well as independent, but this can be relaxed somewhat.)
Also, there's a caveat that the mean and variance (or second moment $E(X^2)$) must exist / be finite. (See e.g. the Cauchy distribution for a problematic distribution.)