Mathematics
Early on I noticed that mathematicians live in a world inaccessible to common mortals... They are a special breed possessed by an intense cerebral life; simultaneously living on two distinct levels of consciousness, they are at once present and able to carry on normally and yet are immersed in the abstractions that form the core of their lives.
In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life.
Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets that word) seems to play a role.
Mathematics is the art of giving the same name to different things.
What is it that a mathematician wants as an artist? I believe that he wishes merely to understand and to create. He wishes to understand, simply, if possible – but in any case to understand; and to create, beautifully, if possible – but in any case to create.
Math research
It's not only the question, but the way you try to solve it.
I don’t have any particular recipe [for developing new proofs]... It is like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out.
What is research? What does research mean? The pursuit of truth? A quest into the unknown? Well, to start, it means finding a problem and solving it. It could be a tiny step of a small puzzle (by standing at the shoulder of a giant!). Or, it could also be a glimpse at a bigger picture (eventually). Since any worthwhile problem won't be easy to solve, the road can be long and windy with twists and turns.
I often heard, "Math is too big. Wait until I have learned enough." The fact is that nobody knows everything. Don't wait. Just get started.
For nine years I studied mathematics in school, sometimes by myself and sometimes following a mentor's suggestions. After school I was supposed and encouraged to do mathematics independently. I set myself the challenge of finding my theorem completely on my own. Finding a new theorem is like finding a gem deep in earth. No one knows where the gem is, and even if one knows where it is, one does not know how to get at it. It is either an almost impossible task or a very exciting and rewarding one, depending on one's ability and attitude toward the gem, mathematics.
Jaigyoung Choe, from My life as a mathematician
Some tips
Here are some tips I find useful for learning things in math (and in life).
What are we talking about?

Organize your knowledge in “boxes”. At the beginning, your boxes will be few and very large. As your experience grows up, your boxes will be smaller and more numerous.

What's the point?

Identify the target and understand it, possibly rephrasing it by using things you already know. Knowing where you want to go is the first step of your journey.

Find a working example.

Make an example using what you already know to see how the new thing works. An abstract and sophisticated statement is not interesting (and even mistaken!) if not worth of a plain and illuminating example.

Where's the beef?
(Peter Sarnak) 
Look for the key idea behind the whole machinery truly unblocking the matter. Good ideas are simple in their substance, easy to be shaped in different forms and extremely powerful in several and seemingly distant situations.

Young man, in mathematics you don't understand things. You just get used to them.
(John von Neumann) 
Question, (re)do and apply what you know. Explore new subjects looking for connections and analogies. Enjoy your achievements  even if small and/or already known  because you conquered them. Playing with your knowledge will make it more and more familiar and precious day by day.
