Suppose that $0$ is a regular value of $f$. Then $0$ is a critical value of $g=f^2$, yet the level set $g^{-1}(0)$ is a regular submanifold. In this case all the points on $g^{-1}(0)$ are critical points of $g$.
The general answer is difficult. You need to assume something about $f$ f$. A natural assumption would be that the critical points of $f$ are isolated and of finite type. Near such points one can find local coordinates so that in these coordinate $f$ looks like a polynomial. (This is a generalization of the classical Morse lemma due to Tougeron.) In such cases you need to understand the zero sets of real polynomials which can be challenging. A nice place toconsult to consult for such issues is the book by Arnold, Gussein-Zade and Varchenko on singularities of differentiable mappings.
