"Mathematical Methods of Classical Mechanics", and "Ordinary differential equations" and "Geometrical Methods in the Theory of Ordinary Differential Equations" by late V.I. Arnold. In my opinion, these are THE books for anyone who wants to understand geometric theory of ODEs. I agree with Andrew's comment though, that the books might be a difficult read for an undergraduate (particularly, the first one).
My pick for Fourier analysis is, well, "Fourier analysis" by T.W. Körner. Very pedagogical, with lots of historical sections and nice illustrations. And it is probably more in the spirit of Needham's book than the books by Arnold.