For a map $f: M \rightarrow N$ between smooth manifolds, the general strategy to show smoothness is to use
Lemma: $f: M \rightarrow N$ is smooth iff for every $x \in M$ there is a chart $(U, \Psi)$ at $x$ in $M$, and a chart $(V, \varphi)$ at $f(x)$ in $N$ such that the map $\Psi^{-1} \circ f \circ \varphi: \mathbb{R}^{dim(M)} \rightarrow \mathbb{R}^{dim(N)}$ is smooth.
So, we look for local coordinate representations of the function $f$ and show that this map is smooth.
How do we recreate this for vector bundles? Vector bundles are defined to be (smooth) manifolds, so it seems natural to try and use the above lemma. However, I'm having trouble unravelling the definitions of local charts on a vector bundle, so I can't even begin to obtain a coordinate representation of any function. My best guess is to use local trivialisations and somehow compose them with the local charts of the base manifolds, but I don't know how to continue.
I am correct in trying to use this lemma, or is there some easier strategy for working with maps between vector bundles?