The situation is, in fact, quite complicated.
- If $G$ is a Lie group, $X$ is a completely regular Hausdorff topological space (e.g. a metrizable space) and $G\times X\to X$ is a Palais-proper (see below) free continuous action, then the projection map $X\to X/G$ is indeed a principal $G$-bundle. This (and much more) is proven in (an unnumbered theorem on page 315, section 4.1)
R. Palais, On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73 (1961) 295-323
although, according to Palais, the result was established earlier by Serre (notes from a Bourbaki seminar). (It takes some time to untangle the terminology used by Palais to understand what his theorem says.) The case when $G$ is a compact Lie group was proven earlier by Gleason.
Here an action is Palais-proper if for every pair of points $x, y\in X$ there exists a pair of their respective neighborhoods $U_x, U_y$ such that the subset
$$
\{g\in G: gU_x \cap U_y \ne \emptyset\}
$$
is relatively compact in $G$. If $X$ is locally compact, this definition is equivalent to the standard one.
- On the other hand, if $G$ is any locally compact metrizable group which is not a Lie group (e.g. the group of $p$-adic integers, which is compact and totally disconnected), then there exists a free Palais-proper continuous $G$-action $G\times X\to X$ on a metrizable space $X$, such that the projection $X\to X/G$ is not a principal fiber bundle. This (which I find quite astounding) is a corollary of Theorem 6 in
S. A. Antonian, Equivariant embeddings into G-AR's, Glasnik Matematički 22 (42) (1987), 503–533.
Namely, Antonian constructs a certain continuous linear actions $G\times B\to B$ on a Banach space and for some vector $v\in B$ with trivial $G$-stabilizer, observes nonexistence of an equivariant retraction of any neighborhood $X\subset B$ of $Gv$ to the orbit $Gv$.
Note that Antonian's examples are never locally compact.
Edit. In fact, there is an example of a compact 1-dimensional Hausdorff space $X$, a free action of a compact metrizable group $G\times X\to X$, such that $X/G$ has dimension 2. In particular, the projection $X\to X/G$ cannot be a locally trivial fiber bundle. The example is mostly due to Kolmogorov
A. Kolmogoroff, Über offene Abbildungen, Ann. of Math. (2) 38 (1937), 36-38
while the version for free action can be found, in English, here,
R. F. Williams, A useful functor and three famous examples in topology.
Trans. Amer. Math. Soc. 106 (1963) 319–329.
In fact, for each $g$, the map $x\mapsto g\cdot x$ is a homeomorphism. Clearly it is continuous and bijective, and its inverse is $x\mapsto g^{-1}\cdot x$, which is also continuous. Therefore in particular the map is open.
To elaborate on why $x\mapsto g\cdot x$ is continuous, the map $f:\{g\}\times X\to X$ is continuous being the restriction of a continuous map, and $x\mapsto g\cdot x$ is $X\to \{g\}\times X\to X$.
Best Answer
$\pi^{-1}(\pi(U))$ is the set of $a\in M$ with $\pi(a)\in\pi(U)$. That is it consists of the $a\in M$ such that $\pi(a)=\pi(b)$ for some $b\in U$. But $\pi(a)=\pi(b)\iff G\cdot a=G\cdot b\iff a\in G\cdot b\iff a=g\cdot b$ for some $g\in G$. Therefore $a\in\pi^{-1}(\pi (U))$ iff there is $g\in G$ and $b\in U$ with $a=g\cdot b$, that is $\pi^{-1}(\pi(U))=\bigcup_{g\in G} g\cdot U$.