How to read a book in mathematics? How is it that you read a mathematics book?
Do you keep a notebook of definitions? What about theorems?
Do you do all the exercises? Focus on or ignore the proofs?

I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.
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)
This is a seven-year post but I am putting here my experience learning mathematics as an undergraduate. 

First, I would explain a bit what's usually inside a mathematics book. For every chapter or sections, there are usually two parts, the contents and the exercises. The purpose of contents is to let you expose yourself to the knowledge while the exercises are to let you improve until you really understand the material well. So, first and foremost, do not feel disappointed if you cannot solve some of the exercises as you have met the exercise at your level! If you can solve all exercises easily or it's so hard that you can't solve any of them, that book is not for you. For me, a book suitable for you would be the one you can solve at least 60% of the questions most of the time. Also, do not try to solve every single exercise in a book as it leads to diminishing returns. There are a lot of fun subjects that you should go for it. This paragraph is to let you have a correct attitude towards exercises in mathematics. Now I will share my way to study a mathematics book. 

The first level - Read the chapters briefly. If you are following a course, lecture's class would be at this level. If you are self-studying, read a chapter that you can understand at least half of the content including proofs. If at this point, it takes so long for you to understand (even the first chapter), consider changing to an easier book. 

The second level - Read the chapters in detail. Jot down the concept or part of proofs that you can't understand. Copy down all the theorem statements and definitions on a piece of paper. This is extremely helpful when you attempt the exercises. 

The third level - Do the exercises in each chapter. After struggling, do not feel embarrassed to look the answer to the questions you cannot solve. Make sure to understand how they solve it. 

The fourth level - After you have completed half of the book. It's time for revision! Try to prove everything in the text once again. Revise the exercise that you had done. If you can't prove some of them, reread again and try to memorise the key parts of the proofs. Then reprove them after a few days. 

The fifth level - First you do the same as in the fourth level to the last half of the book. Then, write a revision sheet for all the chapters. This revision sheet is for your future reference, so it should be short and concise.

Usually, you can read 4 mathematics book in 3 months by following this process provided the level of the book is suitable for you. After this, you may study another subject of your interest or further sharpen your skills by reading a more advanced text of the same subject. For me, it's the fastest way. Of course, to do this, you need to invest at least 8 hours per day on it.