What is the gentlest notes/book of intro differential topology?

I think a combination of Guillemin & Pollack's Differential Topology and Milnor's Topology from the Differential Viewpoint was what I used. There were pros and cons, but if you have a strongish background in multivariable calculus and analysis this could be good.

An Introduction to Manifolds by Loring W. Tu would likely be helpful. It's a textbook written for undergrads, so it works out proofs in detail, but it still covers de Rham cohomology. See Bott and Tu's Differential Forms in Algebraic Topology for a nice sequel when you're done with this one.

As a pretty unconventional suggestion, I would also dare to suggest browsing through Anders Kock's Synthetic Geometry of Manifolds alongside or after reading Tu. Though very different from the usual way of doing differential geometry and topology, it's a great book for getting geometric intuition.

I began with Elementary Differential Topology by James R. Munkres. It has the familiarity of Munkres' style and is rather easy to obtain both on the web and in university libraries.