Book
There is no required text for the course. Notes are available from the lecture notes page. The following books, which I encourage you to consult from time to time, are available from the Math Library.
 [B]

Bachman, David, A Geometric Approach to Differential Forms,
Birkhäuser, 2006.
 [C]

Carmo, Manfredo Perdigão do, Differential forms and
applications, SpringerVerlag, 1994.
Slightly more advanced than this course, with some nice coverage of Riemannian geometry.  [E]

Edwards, Harold M., Advanced calculus, SpringerVerlag, 1996.
One of the earliest undergraduate textbooks covering differential forms. Still recommended as an alternative or supplementary source.  [F]

Flanders, Harley, Differential forms with applications to the
physical sciences, Dover Publications, 1989.
Written for 1960s engineering graduate students, but very concise and lucid (and costs about $10!).  [G]

Guillemin, V. and Pollack, A, Differential topology,
PrenticeHall, 1974.
Written for beginning graduate students, but very intuitive and with lots of interesting applications to topology.  [H]

Hubbard, John H. and Hubbard, Barbara B., Vector calculus, linear
algebra, and differential forms: a unified approach, Prentice
Hall, 1999.
The text for Math 223224 here at Cornell. Covers some of the same ground as Math 321.  [S1]

Spivak, Michael, Calculus on manifolds; a modern approach to
classical theorems of advanced calculus, W. A. Benjamin,
1965.
Efficient and rigorous treatment of many of the topics in this course.  [S2]

Spivak, Michael, A comprehensive introduction to differential
geometry, Publish or Perish, Inc., 1999.
The Great American Undergraduate Differential Geometry textbook, in five massive volumes. It's safe to say few undergraduates (or postgraduates, for that matter) have ever worked their way through all of it, but it reads very well and you may enjoy browsing through it.  [T]

Tu, Loring, An introduction to manifolds, SpringerVerlag,
2008.
 [W]

Weintraub, Steven H., Differential forms: a complement to vector
calculus, Academic Press, 1997.
Written as a companion to multivariable calculus texts, this contains careful and intuitive explanations of several of the ideas covered in this course. Also has a number of straightforward exercises.