Suppose there exists an action of $G$ on $X$. Let $x \in X$ and $H = \text{Stab}(x)$ ($\text{Stab}(x)$ being the stabiliser of $x$). We denote $p_x$ a the canonical projection from $G$ to $G/S_x$.
Let $f_x: G \rightarrow \text{Orb}(x)$, $(\text{Orb}(x))$ being the orbit of $x$ defined as $f_x(g) = g \cdot x$.
My question would be what is the form of a class in $G/H$? Usually it looks like $aH$ with $a \in G$, but here $H$ is equal to $\text{Stab}(x)$. If $g \in aH$, then $\exists\, g' \in G$ such that $g' \cdot x = x$ and $g = ag'$. Is that correct? I am pretty confused over here.
What you wrote is correct. I think the most elegant way to write the left cosets of $H$ in permutational language is as follows:
$$aH = \{ g \in G \mid g \cdot x = a\cdot x\}.$$
(Proving this equality is a good exercise in understanding the definition of a group action.)