Partial derivatives as a basis? Back when I was studying multivariable calculus, I had a sort of "intuition" for the formula for taking directional derivatives (the scalar product of the gradient and the vector that gives the direction). I saw partials as a kind of "basis" of the space of all directional derivatives. Does my intuition serve me right?  Is there a way to make this rigorous? What kind of space (if any) am I talking about?

Your intuition is correct. The concept you have touched on is that of the tangent bundle of a manifold, which is the manifold with the vector space of partial derivatives in one of the point's coordinate charts attached to it at each point. See Wikipedia or any graduate text in differential geometry for details.
