Here are some links to books, lecture notes and survey papers that cover in depth some of the topics mentioned in the blog. Feel free to post suggestions in the comments and we’ll add them to the list.
4-manifolds, Selman Akbulut
A beginner’s guide to Fukaya categories, Denis Auroux
Stein structures: existence and flexibility, Kai Cieliebak and Yakov Eliashberg
Legendrian and transverse knots, John Etnyre
Lectures on contact geometry in low-dimensional topology, John Etnyre (covers (1) foliations and contact structures (2) symplectic handles (3) open book decompositions (4) symplectic caps)
Lectures on open books decompositions and contact structures, John Etnyre
Lectures on knot homology and quantum curves, Sergei Gukov and Ingmar Saberi
Lectures on embedded contact homology, Michael Hutchings
Notes on bordered Heegaard Floer homology, Robert Lipschitz, Peter Ozsvath, Dylan Thurston
Topological methods in 3-dimensional contact geometry, Patrick Massot (convex surface theory)
Higher dimensional contact topology via holomorphic disks, Klaus Niederkruger
A topological introduction to knot contact homology, Lenhard Ng
Geometric decompositions of almost-contact manifods, Francisco Presas
Textbooks:
From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
Introduction to the $h$-Principle
Knots and Links
4-Manifolds and Kirby Calculus 
Monopoles and Three-Manifolds
Introduction to Symplectic Topology
J-Holomorphic Curves and Symplectic Topology
Quantum Fields and Strings: A Course for Mathematicians
Conformal Field Theory and Topology
Chern-Simons Theory, Matrix Models, and Topological Strings
Mirror symmetry, Clay Mathematics monographs
Quantum Mechanics for Mathematicians
Supersymmetry and Supergravity
Introduction to Superanalysis
The Wild World of 4-Manifolds
Notes on Seiberg-Witten Theory, Nicolaescu
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Hello,
Could you add some material for connections to Physics and some beginner’s lecture notes there.
Regards,
Vijay
I don’t know much about connections to physics, except small bits of ad-hoc intuition I’ve occasionally picked up. I found Takhtajan’s book “Quantum Mechanics for Mathematicians” to be really good. But I think connections to physics generally fall outside the scope of this blog.
I am very fan with this kind of learning, i am just asking for you to add
D. Rolfsen, Knots and Links, AMS Chelsea Publishing, AMS (1990)
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