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).

One fruitful way to think about it, if you have physics background, is as phase space. Your manifold is the configuration space for some system of particles, and the cotangent bundle is then the phases, so the cotangent directions are velocities. This is helpful also with the symplectic structure on $T^*M$.