Taylor expansion on Riemannian manifolds

Question

Let $f : \mathcal M \to \mathcal N$ be a smooth map between Riemannian maniolds. Suppose also we have a smooth curve $\gamma : (-\varepsilon, \varepsilon) \to \mathcal M$. Can we say that the Taylor expansion of $f$ around $\gamma(0)$ is given by \begin{align} (f \circ \gamma)(t) &= (f \circ \gamma)(0) + t(f\circ \gamma)'(0) + \frac{t^2}{2}(f \circ \gamma)''(0) + \dots \ ? \end{align} In other words, are the notions of Jacobian and Hessian etc. encoded into the derivatives of $f \circ \gamma$? I suppose you could use the chain rule to expand the derivatives into something like $(f \circ \gamma)'(0) = [ \mathbf J_f(p)]X$, where $\mathbf J_f$ denotes the Jacobian of $f$ and $p = \gamma(0)$ and $X = \gamma'(0)$. Does the above Taylor expansion hold in general for smooth functions and curves on Riemannian manifolds? Also, how does this relate to geodesics and the exponential map?