What is the best book covering basics of topology and geometry for particle physics graduate student who wants to go deeper into gauge theories and string theory? I am looking for something which is not horribly rigorous and verbose.
There are several texts that offer introductions to topology and geometry for physicists, but I feel that a complete education, at least sufficient for physics, requires a pathway through many books. I have outlined my thoughts for what would be best below:
Point Set Topology
Any introductory topology necessitates an understanding of basic point set topology. I personally would recommend the point set topology chapter in Lee's Introduction to Topological Manifolds, as the chapter is very clear and sufficiently comprehensive.
Basic Topology and Algebraic Topology
I think nobody would question Munkres' Topology as a great introduction to the subject, and I find it goes well when paired with Hatcher's Algebraic Topology.
Applications of Topology
Once you have gone through those, and are interested in applying the concepts you've learned in physics, I would recommend Classical Solutions in Quantum Field Theory by Weinberg as well as Nakahara's textbook, Geometry, Topology and Physics for applications; not the maths.
Fibre bundles are ubiquitous in mathematical physics, and a great introduction to fibre bundles is provided by Husemoller's, Fibre Bundles. For a book on understanding the topology of fibre bundles, such as the homotopy theory, especially on a conceptual level, Steenrod's Topology of Fibre Bundles while dated will ensure a thorough understanding.
Smooth Manifolds (Geometry)
Lee's Introduction to Smooth Manifolds provides a sufficiently rigorous but still accessible introduction to the subject, and includes the language of tensor calculus and differential forms, as well as for example the Mayer-Vietoris sequence from algebraic topology to understand cohomology calculations of manifolds.
I personally feel Principles of Algebraic Geometry by Griffiths is a suitable introduction for physicists. This is because it introduces algebraic geometry from initially a more differential geometric perspective, and it provides an introduction to complex manifolds and Riemann surfaces whilst also enhancing your understanding of complex analysis, all of which are essential tools in mathematical physics, most notably in string theory.
Another answer has recommended Nakahara's Topology, Geometry and Physics whilst I have only suggested this for applications. I strongly advise that you do not rely on Nakahara's text for learning the geometry and topology itself, as the book is extremely encyclopaedic and does not work through anything thoroughly, but instead provides rather superficial descriptions before quickly going on to another topic. It's trying to cram too much into one book.
The book by Mikio Nakahara Geometry, Topology and Physics was originally published in a series of Graduate Texts in Physics; there is a new publisher and a new (3rd) edition and I only have the 1st edition. I believe the author wanted to add some material on quantum information in the new edition.
Regardless, you can check out the comments on various sites. The text is advanced in concepts but written with physicists (graduate students or above) as a target audience. It is an excellent starting point.