I am studying The Princeton Companion to Mathematics, but it is heavy, the sections have different difficulties, and time is limited for me. Assuming that I have only some knowledge of undergraduate analysis and algebra but medium mathematical maturity, what are the (most) essential/interesting sections of the book you will recommend to read firstly?
Edit1: Thanks for the current advice by David Quinn. Let me be more specific about this question: What sections of the book are essential readings for preparing to be a math graduate student?
P.S. The table of contents of the book.
It's an amazing book, but it's not the sort of book most people would want to read from cover to cover, but rather to dip into occasionally. Why don't you read the parts most relevant to your course? Or read about something you have never looked at before? It's not exactly a textbook, more of a resource.