I have two differentiable manifolds, $M$ and $N$, and a differential map $f : M \mapsto N$ which is a diffeomorphism. For $X \in \mathfrak{X}(M)$ I want to show a map $f_*X : N \mapsto TN$ is differentiable. The map is given by
$f_*X(q) = (df)_{f^{-1}(p)}(X(f^{-1}(p))$.
Because the map is a diffeomorphism I believe I can claim that $X$ and $f_*X$ are f-related. It follows that I can write, $f_*(X)(h) = X(h \circ f)f^{-1}$ where $h \in C^{\infty}(N)$.
Suppose that $Df:TM\rightarrow TN$ is the derivative of $f$, that is $Df(p, v):= \big(Df(p)\big)(v)$ (or $df_p(v)$ in your notation) where $p \in M$ and $v \in T_pM$. First show that $Df$ is smooth when written in a chart (for this, pay attention to how you define the atlas on the tangent bundle). Then just note that $f_*X$ is the composition $Df \circ X \circ f^{-1}$ of smooth functions.