I'm struggling with a very basic question...could someone help?
Take a smooth manifold $M$, and a curve on it, $\gamma:I \rightarrow M$, where $I$ is an interval of the real numbers. Consider a function on the curve, $f(\gamma)$.
What happens to $f$ if we change the manifold itself? And what if we change the coordinate charts on the manifold?
A curve on a manifold is defined independently of charts. If you have defined it within a chart, you can just use transition maps to go to a different chart.
If we change the (differentiable) manifold, you will need a way to describe the change. This can be done via a smooth map $\phi: N \rightarrow M$. Then the new function will be the pullback of $f$ by $\phi$, defined as $\phi^*f = f \circ \phi$. Now, if $\phi$ is a diffeomorphism, $f \circ \gamma = \phi^*f \circ \phi^{-1} \circ \gamma$. But if there is no diffeomorphism, I don't really see a way to describe the curve in the manifold $N$.