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
http://www.topologywithouttears.net/
This website and its contents should be useful.
If you want to learn some basic algebra, but nothing too in depth take a look at Fraleigh's Abstract Algebra.
For linear algebra , Axler's "Linear Algebra Done Right" is a good introduction.
Best Answer
I note that you wrote complex differential geometry. There is a difference between complex differential and analytic geometry, the latter having much in common with algebro-geometric scheme theory. However, there is no schism between the two - if you're interested in one, you'd better learn a hell of a lot about the other.
For the former, I feel comfortable recommending:
Zheng - Complex differential geometry : A very diffeo-geometrical introduction to the subject. Does not go into extreme technical details, but does not shy away from difficulties.
Demailly - Complex analytic and differential geometry (available for free on Demailly's website) : This is where you'll find all the technical details. Amazing for a second run on the subject, trying but ultimately rewarding on the first. (Note: Demailly recommends Hörmander's book for the complex analytic technical details needed for his own.)
There are books by Werner Ballmann and Andrei Moroianu on Kähler geometry as well. Both are good. Claire Voisin's book on Hodge theory is closely related to what you want, but more algebraic. I imagine you'll also want to look through Shafarevich's books on algebraic geometry, Mumford's books on the same, and anything written by Joe Harris for motivation and examples. Then you'll want to look at Kobayashi-Nomizu as well.
The problem, as always, is that there are seven million different important things to learn. Thus it is pretty much impossible to write a textbook that trails a coherent narrative and covers all these subjects. Also, beware that a "second course" type book on complex differential geometry does, to my knowledge, not exist. The document that comes closes is perhaps Demailly's notes on applications to algebraic geometry, again available on his website.
You should probably read, or rather violently leaf through, all this all at the same time, and then go bother the local complex geometry types with silly questions. There's no better way to learn a new language than to immerse oneself in it and try to talk to the natives.
[edit:] I see I forgot to mention two things: 1. Griffiths and Harris talk about residues in their book, and 2. "quadratic differentials" is secret code for "deformation theory on Riemann surfaces". There is no introduction for beginners to deformation theory. Anyone who says otherwise is either lying or severly underestimating the technical difficulties involved.