Note: I don't have any education whatsoever in logic. This question is supposed to be a mixture of a reference request to sources I can teach myself this sort of knowledge from as well as a question, whether my intuition about this is correct.
I wondered if there is any precise set of rules that I have to follow when writing things down. I think that there are strict rules in logic that one has to follow (which one does intuitively in everyday-mathematics) in order for it to be valid. More precisely, I think there is a strict form of an alphabet which allows to form sentences in maths that are the only sentences I am allowed to write. For example, when writing a statement like (this is just an arbitrary statement without deeper meaning just meant for explanation of the question): Let $f: \mathbf{N} \to \mathbf{R}$ be a function satisfying $$\forall n \in \mathbf{N}: 2f(n)=4.$$ What is happening here? I have an object, called $f$ which denotes a function as well as an object $n$, the natural numbers $2,4$ and the symbols $\in,=,\forall$.
Why can I write this down? I know by axioms that the objects $2$ and $4$ exist and since $f$ is a function, I know that for all $n$ there exists an object in $\mathbf{R}$ calld $f(n)$. Since for any objects in $\mathbf{R}$ we have a product we can write $2f(n)$. Since the symbols "$\in$" and "$=$" are also contained in the alphbet, this should then be a valid sentence, since all the objects that are used exist. This could then be used to form arbitrarely complex statements involving any symbols of the alphabet, as well as objects that exist, such as for example this definition: Let $I \subset R$. A function $f:I \to \mathbf{R}$ is called absolutely continuous on $I$ if for every $\epsilon >0$ there exists a $\delta>0$ such that whenever a finite sequence of pairwise disjoint sub-intervals $(x_k,y_k)$ of $I$ with $x_k < y_k$ satisfies $\sum_k (y_k-x_k) < \delta$, then $\sum_k |f(y_k)-f(x_k)| < \epsilon$ (taken from https://en.wikipedia.org/wiki/Absolute_continuity).
I hope that what I wrote is a somewhat correct explanation and I appreciate any comments as well as possible references, such that I can read more into this.