This is the book we used for the calculus on manifolds class I took this semester. If you respect yourself one bit, you wouldn't use it. Just use Spivak's book on calculus on manifolds instead. This book waters down all the material so much that it is pretty much insulting and the problem sets are soooooooo computational (especially compared to the problem sets in Spivak's book). The sections on differential forms in Munkres are particularly terrible (seriously Munkres what do you have against well formulated rigor and theory?). There are much better books out there for analysis on submanifolds of ##\mathbb{R}^{n}##.
Better yet, since real analysis is a pre-requisitie for analysis on manifolds classes in most cases, you could just learn the necessary topology and do Lee's book on Smooth Manifolds instead.
EDIT: just to give you a taste of the computational annoyance to come if you end up getting Munkres, here is a problem from the chapter on Stokes' theorem: Let ##A = (0,1)^{2}## and ##\alpha:A\rightarrow \mathbb{R}^{3}## be given by ##\alpha(u,v) = (u,v,u^{2} + v^{2} + 1)##. Let ##Y = \alpha(A)##; compute ##\int _{Y_{\alpha}}[x_{2}dx_{2}\wedge dx_{3} + x_{1}x_{3}dx_{1}\wedge dx_{3}]## (the other problems in the chapter are pretty much the same as this one). If you like pointless computations over problems that require proofs then get the book but otherwise I would suggest you stay away from it (I mean he doesn't even write the indices in the correct place for the basis one-forms!).
If you really want to be like Newton, you will not be hurt by practicing computation. I also think most people will appreciate the extra explanation and examples in Munkres. I know when I taught out of Spivak to average math majors at a state school, Spivak was impenetrable to them. Munkres is writing here from years of experience trying to explain this stuff to his many students at MIT. Munkres devotes 380 pages to the topic in comparison to Spivak's 140. This will not be considered a flaw by everyone.
"Obvious" things are not always as obvious as one hopes. E.g. Mike himself corrected one inadequate argument involving partitions of unity in the theory of integration from his first to his second edition, and other subtle flaws that had escaped me have been pointed out by some analyst friends. I agree with you that I myself would probably prefer the Spivak book, but many students might benefit from the fuller version of Munkres.
Indeed after reading the preface to Munkres, I conjecture that he began by teaching from Spivak and gradually wrote his own book to fill in all the missing background he discovered in his classes over the years. This means the book can be expected to all be necessary only to the weakest member of the class, and the others are well advised to select from it what they need. Such a book can serve more people than one that is accessible only to the strongest.
I agree that Spivak is a beautiful book, intended for those who want the briefest possible treatment of the essentials of advanced calculus. (It omits however the theory of existence of solutions of differential equations, unfortunately, as does Munkres also I believe.)
Logged back in after a long period of inactivity solely to respond to WannabeNewton's review of this book. This is a great book for people who have just learned multivariable calculus, which I believe is the intended audience. mathwonk has already outlined its positive attributes. Spivak demands a greater level of mathematical maturity from the reader and is for a different audience even though it covers similar topics. It is not a flaw of the book if the material inside it is too easy for you, it means you're reading the wrong book.