My advice (as a pure mathematician!) is that the best way to get going in research is to write mathematics, and rewrite it till you have got it clear to yourself. The composer Ravel once commented: "Do copy! If you have some originality, it will show itself. If not, do not worry!" In fact the originality may show itself at say the 5th rewrite, when the creaking wheels of the brain have been finally knocked into a little motion.
I did an essay on set theory for a College prize, which helped in an exam question. As a PhD student, I most enjoyed working with a visiting researcher (Dick Swan) on notes of his lectures on sheaves. Much later, I stumbled on the area of groupoids by trying to write a new proof that the fundamental group of the circle was the integers, and writing and rewriting my proof made me see that the arguments should generalise to higher dimensions.
Daniel Quillen was said to have written thousands of pages of notes as he tried to understand various areas, and put them in his own terms.
The question of publication of what you have written is more problematic.
