What you need to know depends largely what you are working on - and that also holds for a successful Ph.D. candidate. Of course, it is good advice to be open-minded beyond one's narrow field of inquiry, as the OP suggests, but a long list of maths topics may not be helpful a lot to beginners.

The Ph.D. period is the time when you have some time available to acquire additional skills, and my advice is: try to strengthen those aspects of your education where you currently have the most glaring deficits. For instance, take a statistics course if that is your weak spot, or learn a foreign language if you have evaded that topic so far in your educational journey.

Read the main textbooks of your field and read and re-read ALL relevant papers for your actual Ph.D. topic, once you've been able to identify it (which may well take you most or all of your first year). Take details notes because nobody can remember most of that much highly concentrated advanced material. Try to find gaps: ask questions and find out if people have tried to answer them yet or not. Interact with others e.g. at conferences, after meeting some people there by attending and networking in a prior year.