Cotangent bundle This may be a poorly phrased question - please let me know of it - but what is the correct way to think of the cotangent bundle? It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object).

I'm not completely sure what you mean by this: "It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object)," but maybe the following will help you see why it is natural to consider the dual space of the tangent bundle.
Given a function f on our manifold, we want to associate something like the gradient of f.  Well, in calculus, what characterized the gradient of a function?  Its the vector field such that when we take its dot product with a vector v at some point p, we get the directional derivative, at p, of f along v.  In a general manifold we don't have a dot product (which is a metric) but we can form a covector field (something which gives an element of the cotangent bundle at any point) such that, when applied to a vector v, we get the directional derivative of f along v.  This covector field is denoted df and is called the exterior derivative of f.
