Is moving into a rotating frame a Galilean transformation? In classical mechanics, we know that laws of physics are invariant in Galilean transformations of the form:
$$ x' = x -vt$$
My question is does shifting to rotating frame also count as a Galilean transformation? If it is not so, then are laws of physics invariant when shifting between rotating frames?
 A: The answers to your questions are no and no.
A full Galilean transformation will consist of a translation and a boost. In 2D (the minimum number of dimensions needed for a rotation, you can write a  Galilean transformation as
$$\left(\begin{matrix}t'\\x'\\y'\end{matrix}\right) = \left(\begin{matrix}t+\Delta t\\ x+\Delta x-v_xt\\y+\Delta y-v_yt\end{matrix}\right)$$
where the coefficients $\Delta t$, $\Delta x$, $\Delta y$, ​$v_x$ and $v_y$ are constant: they do not depend on $x$, $y$ or $t$. While a rotating frame will have something like
$$\left(\begin{matrix}t'\\x'\\y'\end{matrix}\right) = \left(\begin{matrix}t\\ \cos(\omega t)x-\sin(\omega t)y\\\cos(\omega t)y+\sin(\omega t) x\end{matrix}\right).$$
There is no way to put this transformation in the form of a Galilean transformation.
If you go from an inertial frame to a rotating frame (or to any frame not related by a Galilean transformation) then Newton's laws will no longer apply.
There is however a larger framework in which the laws of physics are preserved for arbitrary frame transformations, namely, the theory of general relativity, but this is probably besides the level of the answer you expected.
