I am interested if there is a good classification of finite groups whose characters are all integer valued. One can prove the follow result using Galois Theory:
A group $G$ has the property that $\chi(g) \in \mathbb{Z} \ \forall g\in G,$ and characters $\chi$ of $G$ if and only if $g^{n}$ is conjugate to $g$ for all $g \in G$ and $n \in \mathbb{Z}$ with $n$ and $\operatorname{o}(g)$ coprime.
(where $\operatorname{o}(g)$ denotes the order of $g$ in $G$)
An example of such a group is the symmetric group $S_{n}$ for any $n$.
So my request would be equivalent to asking which finite groups have the property that $g^{n}$ is conjugate to $g$ for all $g \in G$ and $n \in \mathbb{Z}$ with $n$ and $\operatorname{o}(g)$ coprime.
It is unlikely that an answer more explicit than what you have already stated exists. There are many classes of examples of finite groups with this property: Weyl groups (generalizing the symmetric group), Sylow $2$-subgroups of symmetric groups, wreath products of groups satisfying this property with symmetric groups... the list of known examples is enormous! While I am not an expert in character theory of general finite groups, I am vaguely aware that this is a topic of (more or less) current research.
The standard reference for the fact you mentioned, that finite groups with integer valued characters are the same as finite groups in which every element $g$ is conjugate to $g^k$ for all $k$ prime to the order of $g$, is Serre's book Linear representations of finite groups, 13.1.