How to visualize the gradient as a one-form? I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the level sets associated with dual vectors. How to visualize the gradient as a one-form?

To clear up some confusion in the comments: when Carroll refers to the gradient of a function $f$ as a $1$-form he probably intends to refer to the exterior derivative $df$ of $f$. This is a $1$-form containing all of the information which is contained in the gradient, but which can be defined in the absence of a (pseudo-)Riemannian metric. In particular it contains precisely the data needed to differentiate $f$ with respect to any tangent vector; the result tells you how fast $f$ is changing in a particular direction at a particular point, which is what the gradient is supposed to do. 
In the presence of a (pseudo-)Riemannian metric, one can identify $1$-forms and vector fields, and the vector field corresponding to $df$ along this identification is what people usually call the gradient or gradient vector field of $f$. Unlike $df$, it depends on a choice of (pseudo-)Riemannian metric. It's unusual and imprecise to refer to $df$ itself as the gradient. 
