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Especially if your interested in dynamical systems, I highly recommend Abraham--Marsden--Ratiu, Manifolds, tensor analysis, and applications.

For a more Riemannian-geometric/global-analytic focus, you might want to try Klingenberg, Riemannian Geometry, or Lang, Differentiable Manifolds.

There is a standard way to construct a canonical topology on $C^r(M,N)$ for $M$ compact, one that turns $C^r(M, N)$ into a Banach manifold. But I don't think there is a canonical metric on $C^r(M, N)$ unless you put some additional structure on $N$.

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Especially if your interested in dynamical systems, I highly recommend Abraham--Marsden--Ratiu, Manifolds, tensor analysis, and applications.

For a more Riemannian-geometric/global-analytic focus, you might want to try Klingenberg, Riemannian Geometry, or Lang, Differentiable Manifolds.
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