You may read an introduction and outline of 2 Timothy here.
A prayer from Gregory of Nazianus [ (329-389) an early Church father] for your use before reading.
I. 1:1-2 Salutation: Notice how the author refers to Timothy.
II. 1:3-4:8 Body of Letter: Watch for two themes, suffering for the gospel and preserving the tradition.
A. 1:3-2:13 Exhortations to Timothy: Exhortations and also reminders, warnings and Paul’s example.
1. 1:3-5 Thanksgiving: Thanksgiving for Timothy’s faith.
2. 1:6-18 Do Not be Ashamed: A series of exhortations with the common expression, “do not be ashamed”. Shame was a serious concern in the ancient world. The author draws a distinction between shame and suffering. Verses 9-10 are a liturgical fragment. Verses 12 and 18, “that day” refers to the second coming of Christ.
Read More About It
Bassler, Jouette M. “1,2 Timothy and Titus” in HarperCollins Bible Commentary, rev. ed.James L Mays, ed. (New York: Society of Bible Literature) 2000.
Johnson, Luke Timothy, The Writings of the New Testament: An Interpretation, rev. ed. (Minneapolis:Fortress Press) 1999. Chapter 19 “The Pastoral Letters:1 Timothy, 2 Timothy, Titus.
Quanbeck, Warren,Beardslee, William A. “2 Timothy” in The New Oxford Annotated Bible, Bruce M. Metzger, Roland E. Murphy eds. (New York: Oxford University Press) 1994.