I offer that differential geometry may be a much broader field than algebraic topology, and so it is impossible to have textbooks analogous to Switzer or Whitehead.** So, although it isn't precisely an answer to your question, these are the most widely cited differential geometry textbooks according to MathSciNet. I've roughly grouped them by subject area:
- Bridson and Haefliger "Metric spaces of non-positive curvature"
- Burago, Burago, and Ivanov "A course in metric geometry"
- Gromov "Metric structures for Riemannian and non-Riemannian structures"
- Kobayashi and Nomizu "Foundations of differential geometry"
- Lawson and Michelsohn "Spin geometry"
- Besse "Einstein manifolds"
- Abraham and Marsden "Foundations of mechanics"
- Arnold "Mathematical methods of classical mechanics"
- O'Neill "Semi-Riemannian geometry with applications to relativity"
- Wald "General relativity"
- Hawking and Ellis "The large scale structure of spacetime"
- Helgason "Differential geometry, Lie groups, and symmetric spaces"
- Olver "Applications of Lie groups to differential equations"
- Rabinowitz "Minimax methods in critical point theory with applications to differential equations"
- Willem "Minimax theorems"
- Mawhin and Willem "Critical point theory and Hamiltonian systems"
- Katok and Hasselblatt "Introduction to the modern theory of dynamical systems"
- Temam "Infinite-dimensional dynamical systems in mechanics and physics"
- Guckenheimer and Holmes "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields"
- Hale "Asymptotic behavior of dissipative systems"
- Hirsch, Pugh, and Shub "Invariant manifolds"
- Giusti "Minimal surfaces and functions of bounded variation"
Of the metric geometry books (#1), BBI's book is good for self-study, while Gromov's book is nice to have around and open to random pages.
Kobayashi and Nomizu is a hard book, but it is extremely rewarding, and I don't know of any comparable modern book - I would disagree in the extreme with whoever told you to skip it. It is only aged in superficial ways, such as some notations. Lawson and Michelsohn's book is quite advanced, and K-N vol. 1 (at least) would be a prerequisite. It includes a chapter on the Atiyah-Singer index theorem.
Besse's book covers "special Riemannian metrics", including a review of Riemannian, Kahler, and pseudo-Riemannian geometry. It is more of a reference book, good to look through sometimes.
For classical mechanics, Abraham and Marsden is quite sophisticated, and it is necessary to have a solid geometrical footing (roughly K-N vol 1) before going into it; Arnold's book is more introductory and would probably be very nice for self-study.
The general relativity books in #5 are all introductory and pretty approachable.
I'm not so familiar with the books #6-9. Guckenheimer and Holmes seems very friendly.
Personally, I'd also recommend Chow, Lu, and Ni's "Hamilton's Ricci flow," the content of which is necessary to understand the proofs of the Poincare and geometrization conjectures. The first chapter is an excellent mini-textbook on "classical" Riemannian geometry, reaching just beyond introductory books like Do Carmo's.
** just to underline the point in the first sentence - there are only five general or algebraic topology textbooks (Hatcher, Spanier, Rolfsen, Engelking, and Kelley), four differential topology textbooks (Bredon, Hirsch, Milnor "Morse Theory", and Milnor-Stasheff) and two convex geometry textbooks (Schneider and Ziegler) as widely cited as the above differential geometry textbooks
Best Answer
You can take a look at Robert Harper's lectures: http://www.cs.cmu.edu/~rwh/courses/hott/ (There are lecture notes and video recordings) They require much less than you described to understand and covert HoTT.