The actions of $S_n$ and $\mathbb Z_n$ differ in the sense that in the first case the quotient is smooth (it is again $\mathbb C^n$) while in the second case it is singular. This is why in the fist case we have a nice presentation, but in the second not really. For example, the number of generators of the quotient can not be less than the dimension of Zariski tangent space to the singularity at zero of $\mathbb C^n/\mathbb Z_n$.

Still in principle the presentation can be provided by toric geometry (https://dacox.people.amherst.edu/toric.html) because the quotient is the toric singularity. For example, in your case of $\mathbb C^3/\mathbb Z_3$ let us change the coordinates so that $\mathbb Z_3$ is acting as $w_0\to w_0$, $w_1\to \mu w_1$, $w_2\to \mu^2 w_2$ (here $\mu^3=1$). Then you can write the minimal set of four generators:

$w_0, w_1^3, w_2^3, w_1w_2$, and one obvious relation $(w_1^3w_2^3)=(w_1w_2)^3$

The case $\mathbb C^n/\mathbb Z_n$ for $n>3$ will be more involved, but the idea is the same roughly. First you chose the coordinates on $\mathbb C^n$ $w for which the action is diagonal. Then pick the minimal set of monomials (in these new coordinates) that are invariant under the action, and generate the whole set of invariant monomials (of positive degree).

Consider one more case $n=4$, and chose the coordinates $w_i$, so that $Z_4$ is acting as $w_i\to \mu^iw_i$, $\mu^4=1$. The number of generators is $7$ this time:

$w_0, w_1^4, w_3^4, w_2^2, w_1w_3, w_1^2w_2, w_3^2w_2$