Subjects: Don't be ashamed of those who suffer for Christ, Avoid "believers" who persist in sin, "Let everyone who calls on the name of the Lord turn away from wickedness." (2 Timothy 2:19), The Godly will be persecuted by the ungodly, etc.
Paul introduces himself to Timothy, and encourages him saying, "for God did not give us a spirit of cowardice, but rather a spirit of power and of love and of self-discipline." (1 Timothy 1:7) He tells Timothy to not be ashamed of the cross or of Paul's current imprisonment for the faith.
Paul tells Timothy to not deviate from the teachings about Jesus Christ that he has taught. He then speaks briefly about Onesiphorus, who visited Paul in Rome and helped the church in Ephesus.
Paul continues to encourage Timothy as he suffers opposition, giving him instructions on what to tell those who Timothy is instructing. He says to tell them to "avoid profane chatter." (2 Timothy 2:16)
Paul speaks of those who have been "claiming that the resurrection has already taken place," (2 Timothy 2:18) meaning they are saying that Jesus has already resurrected his people, which is false. Paul mentions that he hopes they will eventually "come to know the truth, and...escape from the snare of the devil..." (2 Timothy 2:25-26)
Paul also reminds Timothy, regarding these false teachers, that God's requirements for salvation have not changed, saying "Let everyone who calls on the name of the Lord turn away from wickedness." (2 Timothy 2:19)
He also encourages Timothy to avoid sin and be righteous in 2 Timothy 2:22. Paul says for him to avoid "senseless controversies." (2 Timothy 2:23)
Paul reminds Timothy that people will choose to be sinful, listing all kinds of sinfulness, including narcissism. He tells Timothy to avoid such people.
He also mentions in 2 Timothy 3:7 that there will be those "who are always being instructed and can never arrive at a knowledge of truth." These kinds of people are those who are essentially an infant Christian permanently, never becoming a mature Christian. Paul also notes that these people with "counterfeit faith, also oppose the truth." (2 Timothy 3:8)
Paul mentions how he has lived, as evidence of his godliness, saying, "Indeed, all who want to live a godly life in Christ Jesus will be persecuted." (2 Timothy 3:12) Speaking of scripture, Paul reminds Timothy that all of it is good for learning, teaching, "training in righteousness," etc. (2 Timothy 3:16)
Paul encourages Timothy to keep teaching others, correcting them and encouraging them. Paul explains, "For the time is coming when people will not put up with sound doctrine, but having itching ears, they will accumulate for themselves teachers to suit their own desires, and will turn away from listening to the truth and wander away to myths." (2 Timothy 4:3-4)
This statement that Paul makes has always been true; there are always people who claim to be Christians who will not listen to right teachings, but instead only accept from those who teach them what they want to hear.
Paul tells Timothy to come visit him, sounding urgent about the request, saying "I have finished the race." (2 Timothy 4:7) Paul informs Timothy where each person (who has worked with them) has gone, saying only Luke is with him currently.
Paul also requests that Timothy bring Mark, saying "Get Mark and bring him with you, for he is useful in my ministry." (2 Timothy 4:11) It is interesting that Paul makes this statement, because in Acts 15:37-40 it is recorded that Paul did not initially think highly of Mark.
Prisca (also called Priscilla informally) and Aquila are also mentioned in Acts: more can be read about them in this Wikipedia Article: Priscilla and Aquila.
Paul ends his letter to Timothy, with some final instructions.
Paul writes this letter of 2 Timothy in order to mainly instruct Timothy on how to deal with false teachers, telling him to be confident in rebuking and correcting them.
The main subjects that Paul covers in 2 Timothy are: Don't be ashamed of those who suffer for Christ, Avoid "believers" who persist in sin, "Let everyone who calls on the name of the Lord turn away from wickedness." (2 Timothy 2:19), The Godly will be persecuted by the ungodly, etc.