This book is, of course, designed to be read from first to last page. But other strategies are possible.
Some of Part A (at least Chapter I) should be read before Part B. Some of Part B (at least §§3.3, 3.4 of Chapter 3) should be read before Part C. Within Part B, Chapter 3 should be read first but Chapters 4, 5 and 6 could be covered in any order. Within Part C, the chapters can be read in any order.
A reader familiar with the details of English grammar may prefer to skim over Chapter 2. Note though that §2.7, on complement clauses (which contains some original analysis), should be read before Chapter 7, on complement clauses.
Chapters 4-6 go through every semantic type associated with the class Verb. It is not necessary to study these in detail before looking at some of the discussions of grammatical topics in Part C.