My answer to a question on Quora: Up to what year of mathematics research, does every mathematician need to know all mathematics discovered until that point?
At a practical level: for a beginner mathematician, it is critically important to read classical papers in the field very carefully, line-by-line. With experience, you start scanning papers for key ideas and critical configurations where mistakes are more likely: “aha, the author uses this particular approach; it is interesting to see where and how that particular case is handled in the proof …” This works because an experienced mathematician maintains in his or her mind a mental image of his/her research field and maps the mathematical contents of the paper to that image.
It is impossible to read all publications even in a relatively narrow area of research. You have to be very selective in your reading. Life is short. Mathematics is done by the subconscious; when reading mathematics, you feed your subconscious. It is like feeding a pet — you are responsible for its well-being, and have to maintain a healthy nutritious diet.