I am new to category theory and I have no experience with set theory and logic. I am having trouble with the notion of smallness. In particular, I can not answer to myself questions of the form "Is some category constructed from some other category which is small, small or not?", or "are the hom-sets of some category sets or not?". I will give an example:
Say category $\mathcal{C}$ is small, and given an object $A$ of $\mathcal{C}$ consider the category $\mathcal{C}/A$ of arrows over $A$. Is $\mathcal{C}/A$ small or not?
Well, I know it is, because it is always treated as small in books, but what is the argument? And how much of set theory will I need in order to answer this kind of questions, which are needed for a practical use of category theory? Thanks in advance for any help!
As for your specific question about the comma category $\mathcal{C}/A$, suppose $\mathcal{C}$ is small, that is $Ob(\mathcal{C})$ is a set. Since an object of $\mathcal{C}/A$ is an arrow $f\colon X\to A$ in $\mathcal{C}$, we have that
$$ Ob(\mathcal{C}/A)=\bigcup_{X\in Ob(\mathcal{C})} Hom_{\mathcal{C}}(X,A) $$
which is a set, as it is a union of sets indexed by a set, i.e. it is a union of a set of sets. (I am assuming that each hom-class in your category $\mathcal{C}$ is actually a set, i.e. that you are working with a locally small category).
Now, more generally, if you want to know whether the class $Ob(\mathcal{C})$ of objects of a category is a set or not, you should try to see if you can find one (or more) set $X$ and a finite number of occurences of the axioms of ZF (see here, if needed) which, starting from your set $X$, enable you to assert that $Ob(\mathcal{C})$ is a set. For example, if you know that $X$ is a set and you can recognize that the class of objects of your category $\mathcal{C}$ is, say, $\mathcal{P}(\cup X)$, then you can conclude that it is a set, using the union and the power set axioms.
If you are working within a (fixed) Grothendieck universe $\mathcal{U}$ (so that "small" actually means "$\mathcal{U}-$small"), then the strategy to proving $\mathcal{C}$ small is completely analogous: you need to find some elements of $\mathcal{U}$ and a finite number of applications of the axioms defining a universe $\mathcal{U}$ which allows you to establish smallness of $\mathcal{C}$.
As a quick reference, you may want to take a look at Francis Borceux, Handbook of Categorical Algebra, Volume 1, Paragraph 1.1.