How to read a book in mathematics?
This method has worked well for me (but what works well for one person won't necessarily work well for everyone). I take it in several passes:
Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book.
Read 1: Let your eyes jump from definition to lemma to theorem without reading the proofs in between unless something grabs your attention or bothers you. If the book has exercises, see if you can do the first one of each chapter or section as you go.
Read 2: Read the book but this time read the proofs. But don't worry if you don't get all the details. If some logical jump doesn't make complete sense, feel free to ignore it at your discretion as long as you understand the overall flow of reasoning.
Read 3: Read through the lens of a skeptic. Work through all of the proofs with a fine toothed comb, and ask yourself every question you think of. You should never have to ask yourself "why" you are proving what you are proving at this point, but you have a chance to get the details down.
This approach is well suited to many math textbooks, which seem to be written to read well to people who already understand the subject. Most of the "classic" textbooks are labeled as such because they are comprehensive or well organized, not because they present challenging abstract ideas well to the uninitiated.
(Steps 1-3 are based on a three step heuristic method for writing proofs: convince yourself, convince a friend, convince a skeptic)
From Saharon Shelah, "Classification Theory and the Number of Non-Isomorphic Models"; quoted in Just and Weese, "Discovering Modern Set Theory I":
So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights.
Sorry... I just love that quote.
By accident I came to this question-discussion only today.
The theme of several answers and comments, that many readings in different styles is best, I'd second, at least up to a point.
I would disagree with all advice to refuse to move forward without "mastery of all details prior"... certainly for nearly all textbooks, and even many higher-level monographs. The reasons is that textbooks currently seem to have the style of belaboring every possible detail, in the name of "rigor", as well as being rather sub-verbal about it. That is, the relative significance of different details/lemmas/whatever is not at all delineated. Since at least 90 percent of details are not at all "dangerous", and not even terribly surprising or illuminating, this results in gross inefficiency. Textbooks are 10 times longer than they need to be, and the critical points are lost in a 10-times-larger mess of fussy details. Terrible.
The only serious approach to avoiding drowning in the faux-rigor fussy details is to make at least one pass through material to see the big points, the higher-level plot-arcs. This lends coherence to the lower-level details. "Hindsight" of a sort.
In particular, "exercises" are an extremely volatile issue. Contemporary textbooks "must" include lots-and-lots of exercises to please publishers and meet other expectations. Thus, one has scant idea of the nature of a given one! Also, one can observe the schism in many texts between the "theoretical" nature of the chapter, and "problem-solving" nature of the exercises, with dearth of prototypes in the chapter itself, to maintain a sort of misguided "purity".
So: distinguishing the relative significance of details, and seeing the larger story-arc, are the most important things to cultivate. Some acquaintance with lower-level details is obviously useful, but the purported "ultimate" significance of low-level details is mostly an artifact of the way mathematics is taught in school.