Chapter 1: I’ll note at the outset that the letter is being written during Paul’s (Roman) imprisonment (1:13). Paul welcomes even preaching from bad motives, so long as Christ is proclaimed (1:18). And suffering to advance Christ is a good thing (1:22-24, 29).
Chapter 2: Be unselfish and humble, as Christ was (2:1-8). The universality of Christ is declared (2:1-8). We are told to “work out your salvation in fear and trembling” (2:12). Don’t complain or argue (2:14).
Chapter 3: It is the “true circumcision” that matters, not the old law (3:1-8) — what matters is faith not law (3:9). Note that 3:12-21 makes clear, though, that we are to do our best to live righteously.
Chapter 4: Rejoice in the Lord always (4:4). Be anxious for nothing; pray to be thankful (4:6). Dwell on the good (4:8). Your wealth or poverty shouldn’t matter (4:12). “I can do all things through Him who strengthens me” (4:13). On a personal note, Philippians 4:4-13 is one of my favorite Pauline passages.
My NIV study Bible says the overarching theme of Philippians is joy, and that each of its four chapters develops a different joy theme: Chapter 1 is “joy in suffering,” chapter 2 is “joy in serving,” chapter 3 is “joy in believing,” and chapter 4 is “joy in giving.”