smooth quotient Let $X$ be a smooth projective curve over the field of complex numbers. Assume it comes with an action of $\mu_3$. Could someone explain to me why is the quotient $X/\mu_3$ a smooth curve of genus zero?  

No, the result is not true for genus $1$.  
Consider an elliptic curve $E$ (which has genus one) and a non-zero point $a\in E$ of order three .
    Translation by $a$ is an automorphism $\tau_a:E\to E: x\mapsto x+a$ of order $3$ of the Riemann surface $E$.
It generates a group $G=\langle \tau_a\rangle\cong \mu_3$ of order $3$ acting freely on $E$ and the quotient $E/G$ is an elliptic curve, a smooth curve of genus one and not of genus zero.     
Remarks
a) There are eight choices for $a$, since the  $3$-torsion of $E$ is isomorphic to $\mathbb Z/3\mathbb Z\times \mathbb Z/3\mathbb Z$.
b) The quotient morphism $E\to E/G$ is a non-ramified covering
