Suppose we have a a diffeomorphism between two manifolds, $f: M \rightarrow N$ and a volume form $\Omega$ on $N$. Then is it true that $f^{*}(\Omega) = \Upsilon$ will always be a volume form on $M$?
My thinking is that for any arbitrary vector fields $(X_1, X_2,...,X_N) \in T_pM$ then
$\Upsilon(X_1, X_2,..., X_N) = f^{*}\Omega(X_1, X_2,..., X_N) = \Omega(df_x(X_1),df_x(X_2),...df_x(X_N))$
is always non vanishing given that $\Omega$ is a volume form on $N$? This would be because $\Omega$ is acting on the tangent vectors $df_x(X_i) \in T_{f(p)}N$, which is non-vanishing given that $\Omega$ is a volume form on N.
I'm new to differential geometry, so any mistakes with notation, or comments which aren't accurate, please do let me know!
Your agrument is basically coorect, the only observation you have not put in explicitly that that for a basis $\{X_i\}$ of $T_pM$ the vectors $df_x(X_i)$ form a basis of $T_{f(p)}N$, which is needed to conclude that the value of $\Omega$ on these tangent vectors is non-zero.