For a first year math PhD student, should the student read their graduate level math textbooks page by page, sentence by sentence? Question is: For a first year math PhD student, should the student read their graduate level math textbooks page by page, sentence by sentence?

I often hear professors say that that's a bad idea, and that it's better to look at problems first, and then go back to read as necessary.

From undergrad training though, I like reading math books and working through them methodically, namely, going cover to cover (e.g. for calculus, linear algebra, introductory analysis, probability, etc.)
While it may be good for something, it isn't really the best way to learn anything. To learn, you need to get practice (reinforcement) and feedback. The exercises in the book will do two things for you if the book is a good one. First, they will exercise the more important ideas, giving you the practice, but also giving you an outline of the chapters. Second, the problems will point you, indirectly, to the examples, theorems, and proofs that are the most important to study. 

The big idea here is that not every word in a math book has equal weight. 

Trying to "memorize" the textbook is also a terrible way to prepare for exams. Find more exercises and solve them. 

Your brain isn't like a thumb drive that you can pour information in to an then expect that you have "learned" it. Think of a novel that you read a year ago. How much of it do you really remember? If you read a lot, then it is unlikely that you will have retained very much about that old book. A few people can do that, but they are very rare. Chances are slim that you are one of them. 

Finally, if you can't solve an exercise you know what you need to ask the professor about. Hopefully the prof will give you feedback on your work - the other essential to learning. That way you are less likely to develop misunderstandings that need to be corrected later.