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Answers by Isabelle Guyon, Data Science Prof. Paris-Saclay Univ. (Paris) & President ChaLearn (California), on Quora.
A: There is not just one way. You can start at any age.
Some math background (in linear algebra, statistics, and calculus) is recommended, so take classes on these topics, if possible. Programming is also a useful skill (today people use often Python).
If you are young and want to start it as a hobby, competitions in data science are a great way of engaging (for example from Kaggle). You can then follow classes at the university or on-line (for example from Coursera). Later in your career, if you want to learn about machine learning, I would recommend to start with a project: learn your way by trying to solve a problem that you have at heart. You can find help on the Internet and by attending Meetups. Teachers can tell you what to do, but there is nothing like experience, and nobody will hire you if you do not have any!
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A: To apply machine learning, you can get away with basic first 2 years of university math requirements, preferably notions of linear algebra, statistics, and calculus. But if you want to contribute new machine learning algorithms as a researcher in machine learning, you need to be able to read math books or publications on your own. To get a flavor, browse through the papers of the NIPS or ICML conference. It is harder than when I started because all the low hanging fruits have been picked.
As a graduate student, you will benefit from learning about signal processing, information theory, statistics, and optimization.
Other more specialized classes are sometimes offered including machine learning itself, learning theory, computer vision, robotics, bioinformatics, game theory, neurobiology, artificial intelligence, deep learning, Bayesian networks, causality, etc.
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A: I follow of course the work of the big labs like Facebook and Google. But part of my heart is on the side of causal discovery. I believe that the next big breakthroughs will be in the understanding of how we can make informed decisions using causal models. It is very different to make predictions with the classical I.I.D. hypotheses (independently and identically distributed data) and in cases when you need to predict the consequences of future actions under interventions on a system. What if we administer a certain drug, change the tax law, vaccinate against a particular disease, etc. For making simple diagnoses, causes or effect are predictive (you can diagnose a cold by observing that a patient is coughing), but to cure, you need to understand causes (you need to know whether this is a bacterial infection to administer or not the right antibiotic). So I am looking on the side of the lab of Pearl (UCLA), Glymour, Spirtes, Scheines, and other (CMU), Schoelkopf and his team (Max Planck, Germany), Tsamardinos (Greece), and others.