I decided to do the following problem as an exercise: Let $p \in M$ be a regular point of $f: M \to \mathbb{R}$. Prove the existence of a coordinate system $(x_1,x_2,...,x_n)$ near $p$ such that $f(x)=f(p)+x_{1}.$
Try: Since $p$ is regular, $Df|_{p}$ is surjective and we get a submanifold near $p$ locally homeomorphic to $\mathbb{R}$ (by inverse function, submersion theorems). Can someone hint at the choice of charts so that the restriction to this open neighborhood of $p$ gives $f(x)=f(p)+x_{1}.$ Thank you.
This is more or less the same as the problem at
coordinate system, nonzero vector field
except that the problem there begins with a vector field $X$. That vector field, for you, is just the gradient of $f$. With that starting point, you should be able to turn the rest of the answer into the answer that you need.