Prove that $T_pM^*=\mathfrak m_p/\mathfrak m_p^2$.

Choose local coordinates $x_i$ around $p$, with $x_i(p)=0$. Then $m_p$ is the ideal of functions which vanish at $p$. Such a function is of the form $\sum_{i=1}^na_ix_i+ o(x)$. Then $m_p^2$ is the ideal generated by function which vanishes at the order 2 i.e. $f(x)=o(x)$, and the quotient is just the vector space of linear function $x\to \sum_{i=1}^na_ix_i$, i.e. the cotangent space at $p$.