Palais, "The Foundations of Global Non-linear Analysis" (or his survey article "Homotopy theory of infinite-dimensional manifolds": http://www.sciencedirect.com/science/article/pii/0040938366900024) are handy to have at hand.
EDIT: Also this paper of Eliasson might be useful: "Geometry of manifolds of maps" (1967) Journal of Differential Geometry (available at http://www.intlpress.com/JDG/archive/1967/1-1&2-169.pdf).
Of course, it's always best to see these things in action rather than in the abstract. If you know some differential geometry I can recommend Donaldson & Kronheimer "Geometry of 4-manifolds" (though much of what they do takes place in an affine Hilbert manifold, the lack of generality doesn't make the nonlinear theory significantly easier!) or McDuff & Salamon "J-holomorphic curves and symplectic topology" where they really have used Banach manifolds (for example their universal moduli spaces of pseudoholomorphic curves) and there is a lot of detail on the analysis. Another interesting setting in which infinite-dimensional analysis comes to life is the Ebin-Marsden Annals paper "Groups of diffeomorphisms and the motion of an incompressible fluid" (http://www.jstor.org/pss/1970699) where they do some Riemannian geometry (again in the Hilbert setting, I think).
