First let me say, that I am also trying to gain a understanding of differential forms.
I have found the Geometric Algebra approach for visualizing simple mutlivectors (k-blades) a good way to visualize the infinitesimal multivectors of differential forms.
The first few chapters of Geometric Algebra for Computer Science do a good job of that. http://www.geometricalgebra.net/tour.html
As far as I can tell (only on chapter 4) GA excludes visualizing general multivectors like $a \wedge b + c \wedge d $, but as Dan Piponi pointed out here: http://homepage.mac.com/sigfpe/Mathematics/forms.pdf your probably okay thinking of that construction as two parallelograms.
With those resources as the basis of my visual intuition, the second chapter of Manifolds and Differential Forms by Reyer Sjamaar here:
http://www.math.cornell.edu/~sjamaar/papers/manifold.pdf seemed fairly understandable.