One does not simply learn how to think philosophy and philosophize. Philosophy can be divided into two main aspects: general knowledge (the history of philosophy, the biography of philosophers, and the teaching of philosophical concepts) and creation.
Just like a painter paints and a dancer dances, a philosopher creates and experiments with concepts, ideas, and logical arguments, giving their own unique point of view to the things that exist. Just as there is no one specific way to paint or dance, there is no one specific way to philosophize in the artistic aspect of philosophy, which in turn contributes to the study of philosophy.
There are many ways to philosophize and think philosophically. However, in order for an action to be considered philosophical, it must adhere to a few principles of deep thinking, just as a mass of colors cannot be considered a painting, and just as random movements cannot be considered a dance.
Depth
Philosophy is all about depth and going deeper and more extensively into every possible subject one can consider. Some claim that today's world is a very shallow world, and I can agree with that. We take so many things for granted, and we do not bother asking the big and important questions that define ourselves and our way of life. Depth is all about breaking the limits of the taken-for-granted, in the name of revealing to ourselves and others the non-taken-for-granted, and making ourselves and others think.
Logic
Philosophical thoughts and practices are composed of logical claims, which are contained by arguments. In order to practice philosophy, one needs to form an opinion or a conclusion, and that always needs to come from a rational and consistent process of claims or convincing assumptions, so that the conclusion of the argument you create is more stable in a way that it would be difficult to perform an opposite argument that may threaten your own argument's stability.
Remember that if a philosopher is to be compared to a construction worker, then his logical claims are bricks and the argument is the house that is formed by the said bricks. The more convincing and rational a series of claims will be, the stronger the house will become against possible disasters, which are criticisms. However, as I learned myself in university, not all philosophers can agree to a single claim or argument. Thus, there is no such thing as a perfect argument, and this is because everything is prone to skepticism and criticism.
Skepticism
I claim that philosophy is always formed out of a contradiction, i.e. from some idea or status quo that is opposite to the philosopher's argument. This is because both philosophy and skepticism have one thing in common, and that is to shake an accepted idea or belief by the norms. I believe that the role of the philosopher is to go against the common opinion by forming an opposite opinion in which the consciousness's horizons can grow and evolve beyond the limits of the norm. This is why skepticism is necessary while philosophizing, even in the things which are the most taken for granted.
One can say that philosophy is an evolutionary process where ideas and beliefs are formed on the ruins of the previous ideas and beliefs, and this is what makes the human mind evolve internally - by placing doubt and by searching for a new solution. This happened with Socrates, who wanted to shake the confidence of the people and their knowledge by implying that one knows nothing no matter how assured we are of our expertise in a certain subject, and this happened with Nietzsche who claimed that God is dead and claimed the dysfunctionality of the Christian religion and authority.
Contemplation
The workshop of the philosopher is in solitude, where they can construct and play with ideas and concepts. The tool which the philosopher uses in their craft is their own mind. Kant, for example, believed that every moral principle is able to be recognized and discovered by human intelligence, and other philosophers (Butler, I think?) claimed that our intelligence is a gift from a religious entity.
If you wish to philosophize, you will have to think a lot, and I mean a lot of your time, which also demands massive cognitive effort, especially if you were to attempt to solve a philosophical problem.
Solitude is thus necessary for philosophy because in solitude, one can concentrate beyond the distractions of the daily and of the external life, in the quest to create and to devise something bigger that may, potentially, be of contribution to oneself and to the people who study the philosopher's teachings.
This is also why writing is important and necessary as a form of contemplation: you both think and record your thinking, which may survive dozens if not hundreds and thousands of years after you. Because philosophy is the deepest and yet the most basic of all humanities, recording your thinking and developing it is a must to perhaps make some change in the world.
I remember I read this quote somewhere: If you want to change the world, grab a pen and write. That is the sacred duty of the philosopher: to make bridges for a better world and consciousness shaped by their intellectual craft. Have a good fortune for your important journey!
