Can someone tell me why I read in almost every statistic book (and it's also what I've learned in school) that a sample with 30 iid elements, and with the condition of having some kind of symmetric distribution of the elements can be considered as a "big" sample?
Where does this 30 come from? Why exactly 30?
What a "big" sample is depends entirely on the purpose of your data analysis. A number of $30$ samples is probably big enough that a central limit theorem holds for the mean (even if some outliers or skewness is present), thus a t-Test might be appropriate (you won't have to resort to non-parametric methods such as Wilcoxons test).
Now consider a different scenario, e.g. estimating the parameter $p$ of a binomial distribution $\operatorname{Binom}(n,p)$ where $n = 30$. Then one knows that a $95\%$ confidence set for $p$ is given by $\hat p \pm \frac 1 {\sqrt n}$. So for $n=30$ we have estimated $p$ up to $18\%$ - not very precise.