I recall reading this article that was written to explain how Descartes read philosophy effectively. I am wondering if such analogous tips have been made by past mathematicians?
Mathematicians rarely describe the process which led them to their discoveries. One notable exception was Euler. Some books on the subject written by great mathematicians are:
J. Hadamard, The Mathematician's Mind. The Psychology of Invention in the Mathematical Field (English translation).
G. Polya, a) Mathematics and plausible reasoning. b) Mathematical discovery.
Cédric Villani's Living theorem does exactly that. A great read for scientists, even non-mathematicians.
The first mathematician who did that was Archimedes, in his The Method of Mechanical Theorems (usually known as The Method).
Alexander Grothendieck, without a doubt one of the most creative mathematicians in the last 100 years, wrote a voluminous manuscript Récoltes et Semailles about (among other things) his approach to mathematics. Since he later requested that it be withdrawn from the public, it is hard (but probably not impossible) to find the full text on the internet.
One little snippet of this which has gained some popularity among mathematicians is the "Rising Sea" metaphor. There is an article by Colin McLarty which discusses this, quoting at length translated passages from Récoltes et Semailles.