Consider the action of the cyclic group $G$ of order $n$ acting on an $m$-dimensional vector space $V$. I'll assume you are working over an algebraically closed field $k$ (or at least a field containing the nth roots of unity). Over such a field the elements of the cyclic group are diagonalizable. Thus the group is generated by $g=\diag(\omega^{-a_1},\dots,\omega^{-a_m})$ where $\omega$ is a primitive nth root of unity and the $a_i$ are integers (which may be assumed to lie between 0 and $n-1$.

If $\{x_1,\dots,x_m\}$ is a basis for $V^*$ dual to the diagonal basis for $V$, then $g$ acts via $g\cdot x_i = \omega^{a_i} x_i$. Hence if $t = x_1^{e_1}\cdots x_m^{e_m}$ is a monomial then
$g\cdot t = \omega^{a_1e_1 + \cdots a_me_m} t$. Since a function is invariant iff it is a linear combination of invariant monomials, the invariant theory reduces to solving the linear congruence
$a_1e_1 + \cdots a_me_m \equiv 0 \pmod{n}$ for $(e_1,\dots,e_m)$. These solutions form a monoid in $N^m$. Finding a set of minimal generators for $k[V]^G$ amounts to finding minimal generators for this monoid. This is not really solved in general but there has been a lot of study, as you might imagine.

I suggest the following references.

Weidong Gao, Alfred Geroldinger,
Zero-sum problems in finite abelian groups: A survey,
Expo. Math. 24 (2006) 337 – 369

John C. Harris and David L. Wehlau,
Non-Negative Integer Linear Congruences,
Indagationes Mathematicae {\bf 17} No. 1 (2006) 37-44. arXiv:math/0409489v1

Pingzhi Yuan,
On the index of minimal zero-sum sequences over finite cyclic groups,
Journal of Combinatorial Theory, Series A 114 (2007) 1545–1551

Finally, I'll mention that things are much much more complicated (equivalent to the invariant theory of SL(2,C)) when the characteristic of $k$ divides $n$.

iff$G$ is generated by reflections (linear transformations conjugate to ${\rm diag}(1,1,\ldots,1,-1)$), and the cyclic groups of order $n>2$ do not satisfy this criterion. – Noam D. Elkies Mar 24 '13 at 16:50