Books for general relativity

I can only recommend textbooks because that's what I've used, but here are some suggestions:

Gravity: An Introduction To General Relativity by James Hartle is reasonably good as an introduction, although in order to make the content accessible, he does skip over a lot of mathematical detail. For your purposes, you might consider reading the first few chapters just to get the "big picture" if you find other books to be a bit too much at first.
A First Course in General Relativity by Bernard Schutz is one that I've heard similar things about, but I haven't read it myself.
Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll is one that I've used a bit, and which goes into a slightly higher level of mathematical detail than Hartle. It introduces the basics of differential geometry and uses them to discuss the formulation of tensors, connections, and the metric (and then of course it goes on into the theory itself and applications). It's based on these notes which are available for free.
General Relativity by Robert M. Wald is a classic, though I'm a little embarrassed to admit that I haven't read much of it. From what I know, though, there's certainly no shortage of mathematical detail, and it derives/explains certain principles in different ways from other books, so it can either be a good reference on its own (if you're up for the detail) or a good companion to whatever else you're reading. However it was published back in 1984 and thus doesn't cover a lot of recent developments, e.g. the accelerating expansion of the universe, cosmic censorship, various results in semiclassical gravity and numerical relativity, and so on.
Gravitation by Charles Misner, Kip Thorne, and John Wheeler, is pretty much the authoritative reference on general relativity (to the extent that one exists). It discusses many aspects and applications of the theory in far more mathematical and logical detail than any other book I've seen. (Consequently, it's very thick.) I would recommend having a copy of this around as a reference to go to about specific topics, when you have questions about the explanations in other books, but it's not the kind of thing you'd sit down and read large chunks of at once. It's also worth noting that this dates back to 1973, so it's out of date in the same ways as Wald's book (and more).
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg is another one that I've read a bit of. Honestly I find it a bit hard to follow - just like some of Weinberg's other books, actually - since he gets into such detailed explanations, and it's easy to get bogged down in trying to understand the details and forget about the main point of the argument. Still, this might be another one to go to if you're wondering about the details omitted by other books. This is not as comprehensive as the Misner/Thorne/Wheeler book, though.
A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics by Eric Poisson is a bit beyond the purely introductory level, but it does provide practical guidance on doing certain calculations which is missing from a lot of other books.

This list is extensive, but not exhaustive. I am aware that there are more standard GR books out there such as Hartle and Schutz, but I don’t think these are worth mentioning. Books with stars are, in my opinion, “must have” books. (I) denotes introductory, (IA) denotes advanced introductory, i.e. the text is self-contained but it would be very helpful to have experience with the subject and (A) denotes advanced.

Special Relativity

E. Gourgoulhon (2013), Special Relativity in General Frames. (A) $\star$

This is a rigorous and encyclopedic treatment of special relativity. It contains pretty much everything you will ever need in special relativity, like the Lorentz factor for a rotating, accelerating observer. It is not an introduction, the author does not bother to motivate the Minkowski metric structure at all.

Introductory General Relativity

These books are "introductory" because they assume no knowledge of relativity, special or general. Additionally, they do not require the reader to have any knowledge of topology or geometry.

S. Carroll (2004), Spacetime and Geometry. (I) $\star$

A standard first book in GR. There isn't much to say here, it's an excellent, accessible text that gently introduces differential and Riemannian geometry.

A. Zee (2013), Einstein Gravity in a Nutshell. (I) $\star$

This is one of the best physics books ever written. This can be comfortably read by anyone who knows $F=ma$, vector calculus and some linear algebra. Zee even completely develops the Lagrangian formalism from scratch. The math is not rigorous, Zee focuses on intuition. If you can't handle a book talking about Riemannian geometry without the tangent bundle, or even charts, this isn't for you. It's rather large, but manages to go from $F=ma$ to Kaluza-Klein and Randall-Sundrum by the end. Zee frequently comments on the history or philosophy of physics, and his comments are always welcome. The only weakness is that the coverage of gravitational waves is simply bad. Other than that, simply fantastic. (Less advanced than Carroll.)

Advanced General Relativity

These books either require previous knowledge of relativity or geometry/topology.

Y. Choquet-Bruhat (2009), General Relativity and the Einstein Equations. (A)

A standard reference for the Cauchy problem in GR, written by the mathematician who first proved it is well-posed.

-S.W. Hawking and G.F.R. Ellis (1973), The Large Scale Structure of Space-Time. (A) $\star$

The classic book on spacetime topology and structure. The chapter on geometry is really meant as a reference, not everything is given a proper proof. They present GR axiomatically, this is not the place to learn the basics of the theory. This text greatly expands upon chapters 8 through 12 in Wald, and Wald constantly references this in those chapters. Hence, read after Wald. For mathematicians interested in general relativity, this is a major resource.

P. Joshi (2012), Gravitational Collapse and Spacetime Singularities. (A)

A modern discussion of gravitational collapse for physicists. (That is, it's not a hardcore mathematical physics monograph, but also not handwave city.)

M. Kriele (1999), Spacetime. (IA)

While technically an introduction, because the reader need not know anything about relativity to read this, it's quite mathematically sophisticated.

R. Penrose (1972), Techniques of Differential Topology in Relativity. (A)

This is a proof graveyard. Some of the proofs here are not found anywhere else. If you're willing to skip 70 pages of pure math and take the results on faith, skip this. It overlaps with Hawking & Ellis a lot.

E. Poisson (2007), A Relativist’s Toolkit. (A) $\star$

This is really a toolkit, you're assumed to know basic GR coming in, but will leave with an idea of how to do some of the more complicated computations in GR. Includes a very good introduction to the Hamiltonian formalism in GR (ADM).

R.K. Sachs and H. Wu (1977), General Relativity for Mathematicians. (A)

This is an extremely rigorous text on GR for mathematicians. If you don't know what "let $M$ be a paracompact Hausdorff manifold" means, this isn't for you. They do not explain geometry (Riemannian or otherwise) or topology for you. Put aside the strange notation and (sometimes stupid) comments on physics vs. mathematics and you have a solid text on the mathematical foundations of GR. It would be most helpful to learn GR from a physicist before reading this.

J. Stewart (1991), Advanced General Relativity. (A)

A standard reference for spinor analysis in GR, the Cauchy problem in GR, and Bondi mass.

N. Straumann (2013), General Relativity. (IA) $\star$

A mathematically sophisticated text, thought not as much as Sachs & Wu. The coverage of differential geometry is rather encyclopedic, it's hard to learn it for the first time from here. If you're a mathematician looking for a first GR book, this could be it. Besides the overall "mathematical" presentation, notable features are a discussion of the Lovelock theorem, gravitational lensing, compact objects, post-Newtonian methods, Israel's theorem, derivation of the Kerr metric, black hole thermodynamics and a proof of the positive mass theorem.

R.M. Wald (1984), General Relativity. (IA) $\star$

The standard graduate level introduction to general relativity. Personally, I'm not a fan of the first four chapters, the reader is much better off reading Wald with a basic understanding of GR and geometry. However, the rest of the text is excellent. If you can only read one text in the "advanced" list, it should be Wald. Some topology would be good, the appendix on it is not very extensive.

General Relativity Reference Texts

These are some canonical reference texts.

S. Chandrasekhar (1983), The Mathematical Theory of Black Holes. (A)

Pages and pages of calculations. More pages of calculations. This book has derivations of all black hole solutions, geodesic trajectories, perturbations, and more. Not something you would sit down and read for fun.

C.W. Misner, K.S. Thorne, and J.A. Wheeler (1973), Gravitation. (I)

The most cited text in the field. It is absolutely massive and covers so much. Be warned, it's somewhat out-of-date and the notation is generally terrible. The best use for MTW is to look up a result every now and then, there are better books to learn from.

H. Stephani, et al. (2009), Exact Solutions of Einstein’s Field Equations. (A)

If an exact solution of the Einstein equations was found before 2009, it is in this book and is likely accompanied by a derivation, a sketch of the derivation and some references.

S. Weinberg (1972), Gravitation and Cosmology. (I)

Weinberg takes an interesting philosophical approach to GR in this book, and it's not good for an introduction. It was the standard reference for cosmology in the 70s and 80s, and it's not unheard of to reference Weinberg in 2016.

Riemannian and Pseudo-Riemannian Geometry

Texts focused entirely on the geometry of Riemannian and Pseudo-Riemannian manifolds. These all require knowledge of differential geometry beforehand, save for O'Neil.

J.K. Beem, P.E. Ehrlich, and K.L. Easley (1996), Global Lorentzian Geometry. (A)

A very advanced text on the mathematics of Lorentzian geometry. The reader is assumed to be familiar with Riemannian geometry. Hawking & Ellis, Penrose and O'Neil are crucial, this book builds on the material in those texts (and the authors tend to not repeat proofs that can be found in those three). The sprit of the book is to see how many results from Riemannian geometry have Lorentzian analogues. The actual applications to physics are speculative.

J. Cheeger and D.G. Ebin (1975), Comparison Theorems in Riemannian Geometry. (A)

An advanced text on Riemannian geometry, the authors explore the connection between Riemannian geometry and (algebraic) topology. Many of the concepts and proofs here are used again in Beem and Ehrlich.

M.P. do Carmo (1992), Riemannian Geometry. (I) $\star$

A terrific introduction to Riemannian geometry. The presentation is leisurely, it's a joy to read. Notable topics covered are global theorems like the sphere theorem.

J.M. Lee (1997), Introduction to Riemannian Manifolds. (I)

A standard introduction to Riemannian geometry. When I don't understand a proof in do Carmo or Jost, I look here. It covers somewhat less material than do Carmo, though they are similar in spirit.

J. Jost (2011), Riemannian Geometry and Geometric Analysis. (IA)

An advanced "introduction" to Riemannian geometry that covers PDE methods (for instance, the existence of geodesics on compact manifolds is proved using the heat equation), Hodge theory, vector bundles and connections, Kähler manifolds, spin bundles, Morse theory, Floer homology, and more.

P. Petersen (2016), Riemannian Geometry. (IA)

A standard high-level introduction to Riemannian geometry. The inclusion of topics like holonomy and analytic aspects of the theory is appreciated.

B. O’Neil (1983), Semi-Riemannian Geometry with Applications to Relativity. (I) $\star$

A somewhat standard introduction to Riemannian and pseudo-Riemannian geometry. Covers a surprising amount of material and is quite accessible. The sections on warped products and causality are very good. Since large parts of the book do not fix the signature of the metric, one can reliably lift many results from O'Neil into GR.

Topology

Texts that will elucidate the topological aspects of GR and geometry.

G.E. Bredon (1993), Topology and Geometry. (IA) $\star$

A good introduction to general topology and differential topology if you have a strong analysis background. Most, if not all, theorems of general topology used in GR are contained here. Most of the book is actually algebraic topology, which is not so useful in GR.

V. Guillemin and A. Pollack (1974), Differential Topology. (I)

A standard introduction to differential topology. Some results useful for GR include the Poincare-Hopf theorem and the Jordan-Brouwer theorem.

J. Milnor (1963), Morse Theory.

The classical introduction to Morse theory, which is used explicitly in Beem, Ehrlich & Easley and Cheeger & Ebin and implicitly and Hawking & Ellis and others.

N.E. Steenrod (1951), The Topology of Fiber Bundles.

Most advanced GR books contain the following: "The manifold $M$ admits a Lorentzian metric if and only if (a) $M$ is noncompact, (b) $M$ is compact and $\chi (M)=0$. See Steenrod (1951) for details." This book contains the most fundamental topological theorem of GR, that, to my knowledge, is not proved anywhere else.

Differential Geometry

Texts on general differential geometry.

S. Kobayashi and K. Nomizu (1963), Foundations of Differential Geometry (Vol. 1, 2). (A)

This is the standard reference for connections on principal and vector bundles.

I. Kolar, P.W. Michor, and J. Slovak (1993), Natural Operations in Differential Geometry. (A)

The first three chapters of this text cover manifolds, lie groups, forms, bundles and connections in great detail, with very few proofs omitted. The rest of the book is on functorial differential geometry, and is seriously advanced. That material is not needed for GR.

J.M. Lee (2009), Manifolds and Differential Geometry. (IA)

A somewhat advanced introduction to differential geometry. Connections in vector bundles are explored in depth. Some advanced topics, like the Cartan-Maurer form and sheaves, are touched upon. Chapter 13, on pseudo-Riemannian geometry, is quite extensive.

J.M. Lee (2013), Introduction to Smooth Manifolds. (I) $\star$

A very well-written introduction to general differential geometry that doubles as an encyclopedia for the subject. Most things you need from basic geometry are contained here. Note that connections are not discussed at all.

R.W. Sharpe (1997), Differential Geometry. (A)

An advanced text on the geometry of connections and Cartan geometries. It provides an alternative viewpoint of Riemannian geometry as the unique (modulo an overall constant scale) torsion-free Cartan geometry modeled on Euclidean space.

G. Walschap (2004), Metric Structures in Differential Geometry. (IA)

A very rapid (and difficult) introduction to differential geometry that stresses fiber bundles. Includes an introduction to Riemannian geometry and a lengthy discussion of Chern-Weil theory.

Misc.

S. Abbot (2015), Understanding Analysis. (I)

A gentle introduction to real analysis in a single variable. This is a good text to "get your feet wet" before jumping into advanced texts like Jost's Postmodern Analysis or Bredon's Topology and Geometry.

V.I. Arnold (1989), Mathematical Methods of Classical Mechanics. (IA) $\star$

Look here for an intuitive yet rigorous (the author is Russian) explanation of Lagrangian and Hamiltonian mechanics and differential geometry.

K. Cahill (2013), Physical Mathematics. (I)

This book starts from the basics of linear algebra, and manages to cover a lot of basic math used in physics from a physicist's point of view. A handy reference.

L.C. Evans (2010), Partial Differential Equations.

The standard graduate level introduction to partial differential equations.

J. Jost (2005), Postmodern Analysis. (A)

An advanced analysis text which goes from single-variable calculus to Lebesgue integration, $L^p$ spaces and Sobolev spaces. Contains proofs of theorems such as Picard-Lindelöf, implicit/inverse function and Sobolev embedding, which are ubiquitous in geometry and geometric analysis.

I recommend you those books from the excellent Chicago Physics Bibliography: