Often individuals who claim to be Christians point to a radically-changed life as "proof" of their being real Christians, although they have not been baptized. In response, I refer to Titus 2:11 which says that grace has appeared to everybody and "grace" simply facilitated their change. However, Paul's transition confuses me. He continues to say "we / us". Does this refer to all men, or only to Christians? Would it be accurate to teach that grace has appeared to all human beings? If so, would you agree that it would be accurate to say that non-christians can be "godly" and "upright"? Please help with any thoughts.

Paul is speaking of Christians. After all, they are the ones who look for Christ's return (Titus 2:13), not unbelievers.

Theologians speak of prevenient grace and common grace, which work in the lives of the as-yet unsaved (John 6:44; Acts 16:14). I'd say that non-Christians can be as righteous as Cornelius was (Acts 10) -- yet still are not right with God.

Grace not only brings us to the point of conversion, but also operates in our redemption (Romans 1:7, 3:24) and cleansing (Titus 3:5). It continues to sanctify us (Romans 8:29), and sustains us through hardship (2 Corinthians 12:9; 2 Timothy 2:1). Finally, it is only by the grace of God that we continue to grow (2 Peter 3:18).