In superconductors, is the Meissner effect a result of the surface currents? Basically I’m trying to understand whether the current moving to the surface of the conductor is a cause or an effect of the Meissner effect. If it’s a cause, then what phenomena causes the current to move to the surface?
 A: I don't think it that it is necessarily quite right to think of one of these effects as the "cause" of the other; both of them are consequences of the phase transition that occurs in a superconductor.
However, it is nonetheless possible to say the following.  It is not possible to have the Meissner effect (exclusion of magnetic flux from the interior of a superconductor) without the current being restricted to the surface.  However it is possible to have all current on the surface but the no Meissner effect.  So, in a real sense, the Meissner effect can be see as the more basic phenomenon.
The reason that the Meissner effect entails that there be only surface currents is that the Meissner effect demands that $\vec{B}$ be zero in the interior of the superconductor.  Since, in steady-state configuration, $\vec{\nabla}\times\vec{B}=\mu_{0}\vec{J}$ (Ampere's Law), the field $\vec{B}$ cannot be vanishing in a neighborhood where the current density $\vec{J}$ is nonzero.  Thus a volume current density $\vec{J}$ is forbidden, and all currents flow on the superconductor's surface.
The surface currents produce a magnetic field that precisely cancels any externally applied $\vec{B}$ in the superconducting interior.  However, any other distribution of surface currents will not lead to a vanishing interior $\vec{B}$.  So it is possible to have solely surface currents without magnetic flux exclusion.
