The study for Titus 1 is posted here. In Titus 2, Paul exhorts Titus not only to teach sound doctrine, but to exhort the congregation to godliness at home (Tit 2:3-6) and at work (Tit 2:9-10). Why the emphasis on godly behavior? So that those outside may not blaspheme the Word of God (Tit 2:5), but rather the doctrine of God be adorned (Tit 2:10). Ungodliness does not befit God’s people. When outsiders see God’s people behaving ungodly, the person it speaks ill of first is God and His truth, and then you.
Paul then explains the effect of salvation by grace through faith, and the explanation forms a chiastic structure:
1a) Tit 2:11a, For the grace of God that brings salvation;
1b) Tit 2:11b, Has appeared to all men;
1c) Tit 2:12, Teaching: deny ungodliness + worldly lusts/ live soberly + righteously + godly in the present age;
central axis) Tit 2:13, Looking for the blessed hope + glorious appearing of our great God + Savior Jesus Christ;
2c) Tit 2:14a, Who gave Himself for us, that He might redeem us from every lawless deed;
2b) Tit 2:14b, And purify for Himself His own special people;
2a) Tit 2:14c, Zealous for good works.
Grace toward salvation is the matching pair with zealousness for good works (A pair). Grace is not the opposite or enemy of works, but grace makes goodness and godliness possible. His grace has appeared to all men (as Jesus died for the sins of the whole world) and now God is calling all men everywhere to repent (Act 17:30). Repent of what? Of sin, of lawlessness (disregarding the Torah or Law of God), of ungodliness. The grace of God teaches those who have received it to deny ungodliness and to live soberly, righteously, and godly in the present age (C pair). Those who repent, who receive redemption from sin, who receive the gift of grace to turn around and begin to live in a new way (2 Cor 5:17), are purified to be His own special people out of all men (B pair, compare with Hebrew segullah in Deu 7:1-11).