The quotient map $\pi: X\to X/G$ is a covering map if and only if the action of $G$ is properly discontinuous. In this case, the covering map $\pi$ is regular and $G$ is its group of covering transformations.
The complete theorem is the following:
I already tried almost everything, I just have to prove that the covering map $\pi$ is regular and $G$ is its group of covering transformations but I do not know how to do this, could someone help me please?
Hints: Give the hypotheses, to prove that $\pi$ is regular you need only prove that the action of $Aut(\pi)$ is transitive on the fibers. But what are the fibers ?
To prove that $G\simeq Aut(\pi)$: given $g\in G$ clearly you can associate the map $x\mapsto g\cdot x$ which is an automorphism. Conversely, if $f$ is an automorphism then given $x\in X$, there is $g$ such that $f(x) = g\cdot x$. Show that you can choose this $g$ uniformly locally (i.e. you can find a neighbourhood for which this $g$ works for all points; and for this neighbourhood it's the only possible $g$): you will use the "properly discontinuous" hypothesis. Then "$x\mapsto g$" is locally constant. What does that tell you ?