Invariant polynomials under a group action (hidden GIT)

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:

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

Even without knowing an explicit set of generators, you can compute the Hilbert series with very little work as follows. In general, suppose a finite group $G$ acts on a vector space $V$ over a field $k$ of characteristic not divisible by $|G|$ via an action map $\rho : G \to \text{GL}(V)$. Then $G$ acts on the symmetric algebra $S(V^{\ast})$ (a coordinate-independent description of the polynomial functions on $V$), and the ring of functions on the quotient $V/G$ is the invariant subalgebra

$$S(V^{\ast})^G \cong \bigoplus_{n \ge 0} S^n(V^{\ast})^G.$$

The Hilbert series of this subalgebra can be computed using Molien's formula, which gives

$$\sum_{n \ge 0} t^n \dim S^n(V^{\ast})^G = \frac{1}{|G|} \sum_{g \in G} \frac{1}{\det (1 - t \rho(g))}.$$

In particular the Hilbert series is invariant under extension of scalars (so e.g. the answer doesn't depend on whether we take $k = \mathbb{Q}$ or $k = \mathbb{C}$), which is not completely obvious.

In this case $G = \mathbb{Z}_n$ and $V$ is the regular representation. If an element $g \in G$ has order $d \mid n$, its action in the regular representation consists of $\frac{n}{d}$ cycles of length $d$, so the determinant above is $(1 - t^d)^{n/d}$. Altogether this gives the Hilbert series

$$\frac{1}{n} \sum_{d \mid n} \frac{\varphi(d)}{(1 - t^d)^{n/d}}.$$

I recommend that you read the second chapter of this book https://www.springer.com/gp/book/9783211774168, entitled Algorithms in Invariant Theory. In particular it shows that you you can use Groebner bases to calculate generators for the ring of invariants under the action of a finite group.