For $g > 2$ you're not going to be able to write such invariants as *global* functions on the moduli space.

In general for an algebraic variety $X$, there is a canonical morphism $a \colon X \to \mathrm{Spec}(H^0(X,\mathcal O_X))$, the *affinization* of $X$. Points $x$ and $y$ of $X$ are separated by some global regular function if only if $a(x) \neq a(y)$. So your question can only have a positive answer if the affinization morphism is injective (so $M_g$ is affine or quasi-affine). Now $M_2$ is affine, but $M_g$ for $g >2$ contains a complete curve. Since a regular function restricted to a complete subvariety is constant, this means that $a$ has positive dimensional fibers. In fact one can prove using the Satake compactification that through any two points of $M_g$ passes a complete curve, if $g>2$. So the affinization is in fact a point.

This is also why in the $g=3$ case of David Lehavi's answer there are only such choices of invariants on the hyperelliptic locus and on the non-hyperelliptic locus separately. Both loci are affine, but their union is not.

It also seems relevant to mention that $M_g$ is of general type for $g \gg 0$, so that there will be no rational parametrization of the set of genus $g$ curves. And it also seems relevant to mention Fenchel--Nielsen coordinates: these will give you local real-analytic coordinates around any point of the moduli stack in any genus.