The big idea is that for someone who is new to math research, there is no guidance available to solve problems (not exercises in texts but true math problems....there is a difference.....problems can be solved only through research while exercises are meant to be solved by simply understanding the concepts well.....) in math and this book is a great substitute. For example, contest problems like that in the Olympiads are really hard. Traditionally, most trainers in the Olympiads teach the kids the various "tricks" involved in solving contest problems; however, if the kid has not seen a particular type of trick to be used in a problem, he simply fails. This has also been admitted by Arthur Engel in his preface to the fantastic problem solving book that he has written. This book (i.e. Thinking Mathematically) does not talk about tricks. As far as I am concerned, this is the REAL deal. After you graduate from that phase of life where you are done with Olympiads and Putnams, you will encounter a phase where people are actively trying to figure out facts about twin prime conjectures etc. No trick is going to help you there. Professors usually guide the student when it comes to problem solving in pure math. They talk about how to break the problem down, look at it in different angles, understand the logic in the solution to a parallel problem etc..... For a novice who is doing self study, the ONLY other book close to this one is Poyla's Mathematics and plausible reasoning. That seems to be a great book but is more detailed and speaks about induction and all that. He does speak a little bit about specialization and generalization. His other book titled How to Solve it is useless according to me.
Start from this book. There is no doubt that this is THE BEST ONE available when it comes to understanding the ways to crack hard problems or to go about analyzing and investigating them. The author says that it is important to "feel" the emotions while trying problems. You must bring into your cognition as to WHAT EXACTLY HAPPENED when you solved the problem or when you make progress. What clues did you succeed in getting and what did you miss. Keep track of all that and have a picture in your mind about your previous experiences about what kind you clues you typically fail to guess etc. He helps us VISUALIZE and REALIZE the process of investigating a problem in order to get a solution.
Research in pure math is truly speaking, a process of investigation. You have to spend time to come up with those chain of logical sentences which when connected gives you the proof for a Theorem. But in order to arrive at that, you need to investigate. This book is about the process of investigation. If you are really serious about doing research in math, get this book and read it VERY SERIOUSLY and multiple times. I read it the first time and felt like "ok .... I get it...." but when I tried to use it on solving a tough problem, I realized that I did not understand the process of investigation given in this book. I read it a few more times KEEPING IN MIND the kind of problems that I was attempting to solve and I swear it HELPS LIKE NO OTHER BOOK CAN. Be it a kid trying for math Olympiad or a budding researcher, you've gotta get this book. Now if you have a phenomenal professor who has Field's Medal and all that, then I don't know if you really need this book. But in any event, this is THE book to UNDERSTAND THE PROCESS OF INVESTIGATION in solving problems pertaining to pure math. It does have a lot of interesting puzzles and problems but they ARE NOT SUPER HARD. I strongly suggest that you try to apply these principles to solving the Olympiad problems in Arthur Engel's book on math Olympiad problems. Even though his strategy of "try for an hour and look at the solution" is the way to go if you have to solve all the thousands of problems in that book to prepare for Olympiad and learn the tricks properly, I suggest that you try the investigation mathod proposed in this book if you are not preparing for Olympiad so that you truly get a feel for what the authors are saying in this book titled Thinking Mathematically.
Added the following on April 14th of 2014.
There was a question that I was trying. It required me to prove that the harmonic series is never an integer. That is given a positive n greater than 1, the sum 1+1/2+1/3+....+1/n is never an integer. I was about to give up since I did not make a breakthrough in the problem. There is a chapter on "Still Struck" in this book that I read. Great chapter. I read that over and over with this problem in mind. Admittedly, I was about to give up after one week. Then I simply made a list of ideas that I had tried.
1. Let that sum corresponding integer n to be pn/qn where gcd(pn,qn)=1. Can I show that qn>1 for any n>1?
2. Can I show that one of pn, qn is always even and the other is obviously always odd?
3. Can I look at the algorithm to find the gcd to make an inference about pn and qn?
4. Can I assume that pn/qn = an integer for some n and arrive at a contradiction?
All these questions did not lead me anywhere directly. I needed a substantial fact to prove the statement. Strangely, I remembered this thing about harmonic series diverging. There, if 2^m <= k <2^(m+1) where k appears somewhere in the denominator of the harmonic series (hence 1 < k <=n), then they used the idea in the argument that 1/k is always greater than 1/2^(m+1). This made me think let me look at the powers of 2 in qn. Presto! Another deus ex machine. For a qn where n is such that 2^m <= qn < 2^(m+1), the highest power of 2 in qn when pn/qn is in its most reduced form (hence gcd=1) is 2^m!! That's it. If I could make this rigorous, then 2. above follows directly from this.
The thing is that tough problems are insight problems. If we need to show that statement A implies statement B or if we need to find something, then in either case, there are certain givens in the problem. Along with the givens, we need to use data from our memory to LOOK at as many facts as possible. The implication or the thing to be found will follow from ONE OF THOSE OBSERVATIONS. The problem will become harder if more than one thing has to be found. This is called the crux move of the problem as described by Paul Zeitz in his famous problem solving book.
Another crux move that could solve this problem is by making use of the fact that between n and 2n, there is always a prime.
All in all, I have realized that solving tough math problem is about discovery. Cognition fails me initially when I read a tough problem. Then with specialization and generalization on the problem and by mulling over the problem, it is possible to solve atleast some of them if not all.
I have been looking at another problem right now. Let f(n) be the sum of the digits of a number n. So f(1232)=8. What is f(f(f(4444^4444)))? This is an IMO problem.
I feel much better now when I look at such problems. I do not get struck the way I used to before. Before, I would be like a deer in headlights. Simply no idea what to do. Then look at the solution and go "this was so easy!!". Now I know that the solution will be based on a few observations and I have to MAKE THOSE OBSERVATIONS. I may or may not solve it. But there is a difference between the past and the present state of mind when I look at such problems. Before, I did not know what to do and as to how such problems are solved. Now I have the CONVICTION that it is about discovery of certain facts which will surely exist and will lead me to the solution. I may not solve all the problems. But I am glad that I developed the CONVICTION about the insight problems after I read Thinking Mathematically.