You, however, have followed my teaching, my conduct, my aim in life, my faith, my patience, my love, my steadfastness, my persecutions and sufferings that happened to me at Antioch, at Iconium, and at Lystra—which persecutions I endured; yet from them all the Lord rescued me. Indeed, all who desire to live a godly life in Christ Jesus will be persecuted, while evil people and impostors will go on from bad to worse, deceiving and being deceived. But as for you, continue in what you have learned and have firmly believed, knowing from whom you learned it and how from childhood you have been acquainted with the sacred writings, which are able to make you wise for salvation through faith in Christ Jesus. All Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness, that the man of God may be complete, equipped for every good work.2 Timothy 3:10-17
In contrast to these people who Paul describes as becoming "lovers of self, lovers of money..." among other things, the focus is shifted to Timothy. Timothy has followed Paul and all that he's about. Timothy is doing well and Paul encourages him to "continue in what you have learned and have firmly believed". With regards to continuing in what he has learned and believed, Paul brings to mind who he learned from. Paul has just gone into detail about himself and the things about him that Timothy has followed, so he is probably including himself in this. However, there is a note that says "whom" is plural and in the Greek. So, there are probably multiple people he is referring to. Perhaps he is also referring to his mother and grandmother who were both mentioned back in chapter 1. In any case, he is called to remember those people. These people are called to memory to point to the truth and trustworthiness of what he believes. Those people taught Timothy and on the basis of their reputation, Timothy is to remember and trust in what he believes. This makes sense when seeing the many things Paul speaks about himself, since it will remind Timothy of Paul and what Timothy has seen Paul experience. The many experiences will show what kind of person and character Paul is and hopefully be a strong case for the trustworthiness of his teaching. I can remember with great fondness the times shared with some of the people who taught me. As I think about the lives of people who taught me, I am reminded of their character and it encourages me towards remaining faithful to the truths I learned from them.
The second thing brought to attention is the sacred writings. What Timothy has learned and believed is not only based on people, but on scripture. It is the words of God, which should certainly carry some weight to say the least. If trust can be placed on teaching on the basis of a person's conduct and character, the words of God can certainly be trusted on the basis of God and his character. In any case, there are a number of purposes for scripture that Paul lays out. For one, it's "able to make you wise for salvation through faith in Christ Jesus". I don't think this means that salvation is only possible through the learning of the scriptures as a child. Instead, Paul is pointing out the truths taught through scripture to help lead one to salvation. It's not as if "thou must knowest scripture before thou can be saved". I'm sure that there were many who were saved who didn't know any bible at all. Sure, one who is being preached the gospel is being taught what the scriptures say, but it's not as if they need a degree in bible before they can be saved. In any case, scripture is also "for teaching, for reproof, for correction, and for training in righteousness". I think it's easy to see that scripture is for teaching. We are taught by scripture and certainly learn things. We are also shown sin in our lives because of scripture. Scripture shows us what is wrong and shows us that we are sinful and that we do sin. Sometimes it's sharp, like a sword and pierces us in ways nothing else can. It's also useful for correcting. I may be misguided about something and through the study of scripture I am shown the right way as opposed to the wrong way. Sure, I may have my own ideas about a particular thing, but I know that God's word is true and to trust in it would help reveal what is true and what is false. In any case, scripture has many purposes and scripture is certainly good. These things are all for the completion and equipping of a person for good works. To be aligned with God's word is to be prepared for the work God prepares for his people.