As one of my grad school professors used to say, the proper way to read a math book is with pencil in hand. If you don't completely understand something, take the time to work it out. Don't settle for good enough, either--dot all the i's, cross all the t's, and get the details right.

The suggestion to take material out of order is a bit of a mixed bag. A typical book will have a few core chapters, and some extra topics later on. Most chapters will follow a similar outline. You can skip the optional chapters and the optional sections within chapters, but if you skip chapter 2, you probably won't be able to follow much of chapter 3.

One last comment: there are huge variations in how well different books are suited for self-study. Rudin's "Principles of Mathematical Analysis" is a collection of true statements with zero motivation or exposition, and it's impossible to follow on your own. On the other end, something like Herstein's "Topics in Algebra" clearly spells out the reason for everything and is a joy to read. Most books are somewhere in between, so if a certain book isn't working for you, try something else on the same topic.