I've been playing around with Riemann surfaces of cubics, and it seems to me that all coverings of the Riemann sphere from equations of the form
$w^3 = q(z)$, where $q(z)$ is a cubic with three distinct roots, must be isomorphic. Is this correct? (Argument given below.) 


Mainly I want to know if there are any good references on this specific kind of cover.


Argument:
we have a critical point of multiplicity 3 at each of the roots of $q(z)$. Monodromy around a small counterclockwise circuit about any of these points multiplies $w$ by the same cube root of unity ($\neq 1$). So monodromy around a circuit enclosing all three roots of $q(z)$ leaves $w$ unchanged, so no branch points over $\infty$.


Now we can move the three roots of $q(z)$ to any other positions using a Möbius transformation, so by the previous paragraph, we should be able to establish an analytic isomorphism between the surfaces via continuation.



Riemann surfaces of $w^3 = (z-a)(z-b)(z-c)$
Yes, you can view this equation as a family of curves (cyclic covers of $\mathbb{P}^{1}$) , as you said we can move each three points to $0,1,\infty$ and hence your family in fact reduces to a point in the moduli space of curves (or $A\_{g}$). Your curve will have genus $1$ and is smooth so is an elliptic curve. You can find alot about cyclic covers of $\mathbb{P}^{1}$ in the book "cyclic coverings, calabi-Yau manifolds and complex multiplication" by J.C.Rohde. The main stream of this book as the name suggests is something else, but the generalities on cyclic covers is treated quite nicely here.


