Let me tell you that the only thing that I have been doing for the last four years of my life is mathematics. I have enjoyed the experience thoroughly but I have also had points where I was somewhat unsure as to how to approach my learning. I think that there is no one rule that works for everyone; however, let me answer some of your questions. I hope that I can help:

Question: How to study mathematics the right way?

Answer: I think that the best way to study mathematics is as follows. Let us assume that you have already chosen a mathematics book on a subject that you are really interested to learn. When you read the book, aim to actively think about the subject matter in different ways. For example, if a definition is presented, spend at least 30 minutes to think about the definition. If you are studying a book on linear algebra and the definition of a "nilpotent operator" is presented, you should try to discover some basic properties about nilpotent operators on your own without reading further. This can be difficult at first but ultimately an ability to do this effectively with as many definitions as possible is important in research mathematics.

Let us take the following example in elementary group theory. The author presents the definition of a maximal subgroup of a finite group $G$: a subgroup $M$ of $G$ is said to be a maximal subgroup if $M$ is a proper subgroup of $G$ and if there are no proper subgroups of $G$ strictly containing $M$. You should try to take the following steps:

(1) Find examples of maximal subgroups in finite groups and begin with the most trivial examples! For example, the trivial group can have no maximal subgroup. If you understand this, you have grasped one point of the definition. The next step is to consider the simplest cyclic groups. What are the maximal subgroup(s) of the cyclic group of order 2? What are the maximal subgroup(s) of the cyclic group of order 4? Think about basic examples such as this one. When you are ready, try to formulate a general theorem on your own which concerns maximal subgroups of a cyclic group of order $n$. You should arrive at the theorem that a subgroup $H$ of a cyclic group $G$ is maximal if and only if the number $\frac{\left|G\right|}{\left|H\right|}$ is prime.

Continue to find other examples of maximal subgroups in a finite group. The next step is to consider the Klein 4-group and the permutation groups of low orders. I hope at this point you are really fascinated by the concept of a maximal subgroup. At first, the definition might seem like something arbitrary; however, now that you have thought about it, you have started to gain a sense of "ownership" over the definition.

(2) It is now time to formulate and prove some theorems about maximal subgroups. Again, think of the easiest examples. One thing that can be discouraging for a beginner is to not be able to answer a question that looks easy over a long period of time. What is a good example of an easy theorem? You can study those finite groups which have exactly one maximal subgroup. What can you deduce about such a group? If you find that you are stuck, try to work back to the examples of maximal subgroups that you devised earlier. In fact, this question can be answered quite satisfactorily; a finite group with a unique maximal subgroup is cyclic of prime power order.

(3) The next step is to conjecture some more properties about maximal subgroups based on the examples you devised in (1). For example, you worked out that if $H$ is a maximal subgroup of a finite cyclic group $G$, then $\frac{\left|G\right|}{\left|H\right|}$ is a prime number. Is this true for all groups $G$? Can you think of groups $G$ for which this is true?

Notice how one can deconstruct a simple definition to arrive at a host of interesting questions? This is what a mathematician does all the time and is a very important skill. It might seem difficult at first but doing this will make mathematics all the more exciting and will give you a sense of "ownership" over the content. You worked out this piece of mathematics. This is the way I learn mathematics and I can tell you with confidence that if you practice this, it will soon become the norm.

What do you do after you look at the definition and have thought about it extensively? You continue reading the text. There is a good chance that you will notice the author stating some of the results that you discovered on your own. With luck, there will be results that the author has not stated. If this is the case, it could be a good idea to ask (on this website, for example) about the originality of the result.

However, you will encounter theorems concerning the definitions that you simply did not think about. You should resist the temptation to see the proofs of these theorems and rather you should try to prove these theorems on your own. Think about the theorem for at least a few hours before giving up. Note that theorems with quite short proofs can require highly original ideas and therefore you should not pressure yourself to prove the theorem in a small amount of time.

At first, you will take a long time to prove some theorems. There will be routine theorems and these should be proven fairly quickly. But there will also be difficult theorems. As you become experienced, your thinking will be faster and these theorems will come more easily to you. However, you should not expect this to be the case initially.

For example, you might encounter the following theorem in linear algebra: if $N$ is a nilpotent linear transformation from a vector space $V$ to itself and if the dimension of $V$ is $n$, then $N^n=0$. Working out how to prove this theorem on your own is a very valuable and rewarding experience. If you have not seen it already, I suggest that you try to prove it. It is not too difficult, however.

Question: How to avoid forgetting mathematics?

Answer: I used to forget mathematics too when I learnt it. I have talked to various mathematicians about this and they have said exactly the same thing. The point is that you just have to accept from the start that you will forget what you learn. However, there are ways to ensure that you keep this to a minimum.

For example, the best way to not worry too much about forgetting mathematics is to work out the mathematics on your own. For example, consider the steps that I suggested in the previous question. Even if you do this, you can still forget the mathematics, especially if the result in question was fairly easy to prove. (Note, however, that if the result is hard to prove, and you spend, let us assume, 10 hours to prove it, then you will probably never forget it for the rest of your life.)

The best method to take is to write down all the mathematics that you learn. Take copious notes. For example, when I read Walter Rudin's "Real and Complex Analysis" last year, I took down 3 entire books of notes. In fact, I wrote down 600 pages of mathematics when I only read 315 pages!

Write down every definition, every theorem, and every proof. The definitions and theorems should be produced verbatim from the book since it is important to ensure that your understanding of the rigor is correct. However, the proofs should be written in your own words.

Question: How to have a healthy lifestyle?

Answer: I am afraid I really do not have a good answer for this. In the four years that I have been studying mathematics, I have certainly not done anything else. Therefore, I cannot really give advice on how to manage one's time. If you are a serious student in mathematics, you will find yourself spending virtually your entire day doing the subject. This is inevitable. For example, I set myself goals every day of how much mathematics I wish to do and usually I end up doing mathematics non-stop. Nonetheless, I really enjoy this and I would not wish to have it any other way.

But I can offer one small piece of advice: try to wake up early, let us assume, at 6:00 AM. However, do ensure that you sleep for at least 8 hours; therefore, go to bed at 9:00 PM. Sleep is one of the most important points when it comes to studying. Over many years of doing mathematics, I have found that I am most productive and energetic before 12:00. If you can finish off most of your work before 12:00, then you will be in a really good position to do well each day. Also, try to avoid eating big meals. Big meals often cause you to lose your concentration and this can, in turn, lead to several wasted hours.

I think the most important point when you set out to achieve any goal in your life is to take it day by day, hour by hour, even minute by minute. Often you can complicate goals too much by thinking of what you would like to do over the next 1 year or even one month. If you work hard each and every day and set realistic goals, then anything should be possible.