Paul writes to Timothy in Ephesus, warning of legalists and reiterating God's gace and mercy in saving sinners, especially Paul himself.
What can I learn from this?
Having just studied Titus, I know that Paul does not think that Christians should behave however they please; rather we should do good, behaving in a manner that pleases God. However, in this letter Paul is warning about the dangers of false doctrines and meaningless talk (cf Titus 3:9), which he details as (Jewish) myths and endless geneaologies - presumably this was arguments over who was more important among the believers because of their relationships, perhaps who told them the gospel or who baptised them (cf 1 Corinthians 1:11-17).
In contrast to the legalists, Paul wants Christians to focus on what they believe and to have faith. In this chapter, Paul uses the word faith 7 times. Paul writes of Timothy being his son in the faith (v2), that God's work is by faith (v4), that the goal of Paul's commands is love which comes from a sincere faith (v5), that Christ Jesus considers Paul to be faithful (v12), that grace, faith and love were given abundantly to Paul in Christ Jesus (14), and finally that Timothy should hold on to faith (v19a) so his faith may not be shipwrecked, as that of others has been (v19b).
Two other key words in this passage are grace (2 times) and mercy (3 times). From 1:13-16: "... I was shown mercy... The grace of our Lord was poured out on me abundantly, along with the faith and love that are in Christ Jesus. Here is a trustworthy saying that deserves full acceptance: Christ Jesus came into the world to save sinners - of whom I am the worst. But for that very reason I was shown mercy so that in me, the worst of sinners, Christ Jesus might display his unlimited patience as an example for those who would believe on him and receive eternal life."
How can I apply this to my own life?
It is God's mercy to me (that is, the withholding of his righteous judgement and deserved punishment - another word with similar meaning is clemency), and his grace (that is, his undeserved favour or free gift of salvation through faith in Jesus Christ) that allows me to state with confidence, "Christ Jesus saved me." I must hold onto my faith, continuing to belief in the gospel of Jesus Christ, acting on that belief in all that I do, every day.