This is a question that involves lots of research, so don’t be surprised if everyone doesn’t jump up with an answer right away.
You’ve hit the big ones with Peter Paul and John. Paul’s successors in the Bible include Timothy and Titus, and he founded the Church in Greece during his missionary travels as recorded in the Acts of the Apostles. One of the Apostolic Fathers who is associated with him is St. Thecla of Iconium, who is notable as the only female Apostolic Father (or Mother I suppose) and one of the earliest witnesses to ecclesiastical monastic life. The Acts of Paul and Thecla, written in about 160 A.D. and one of the earliest “lives of the saints,” mentions that she “led a monastic life” with several other women after being converted by St. Paul and interacting with him for several weeks. (source) She is also known for being an early witness to Purgatory and the possibility of salvation for non-Catholics. (She saw a pagan woman’s daughter in purgatory and told her to pray for her so that she could enter heaven. The pagan woman had not known about Christianity previously so her daughter could fall under invincible ignorance.)
Another Apostolic Father associated with St. Paul is St. Barnabas. He may or may not be the one mentioned throughout Acts, but it is my understanding that he has always been understood as a disciple of St. Paul. The Epistle of Barnabas is counted among the Apostolic Fathers and is an early witness to the Trinity, the Deuterocanonical books (it quotes from Wisdom right beside Isaiah), opposition to abortion and contraception, the possibility of sacred images, the Catholic view of baptism, the sacrament of confession, and worshiping on Sunday.
St. Peter, of course, ordained St. Clement, but there are several others associated with him as well: Hermas, who wrote the Shepherd of Hermas, was aquainted with Rome and mentions the duties of the pope. As a first or early second-century book, it is also noteworthy because it talks about one of Peter’s successors, and doctrines such as the Trinity, creation out of nothing, offering up penances, mortal and venial sins, the Catholic view of baptism, the permanence of marriage, the intercession of saints, the possibility of private revelation, and the Catholic view of angels and demons. St. Linus is another early saint associated with St. Peter. We don’t have any writings by him that I’m aware of, but his acts are described in the Liber Pontificalis, and he knew St. Peter.
St. John of course taught St. Polycarp, St. Ignatius of Antioch, and St. Papias. We have writings by all of them, though Papias is only in fragments. He attests that we must follow Tradition and not only Scripture. He is also a possible early witness that St. James was not the literal son of Mary, since he possibly mentions that he was the son of “Mary the wife of Cleophas or Alphæus,” “the sister of Mary the mother of the Lord.” (source)
St. Ignatius is known for being an early witness to the Trinity, the divinity of Jesus, the name “Catholic,” the papacy as coming from Peter in Rome and maintaining leadership after his death, the existence of early religious orders, offering up penances, mortal and venial sin, the Catholic view of Baptism, the real presence of Jesus in the Eucharist, the Mass as a sacrifice, the sacrament of confession, the distinctions between priests, deacons, and bishops, worshiping on Sunday, and the merit of good works.
St. Polycarp, meanwhile, attests to holy days, the name “Catholic,” the Deuterocanonical books (he quotes from Tobit right next to Isaiah and 1 Peter), the resurrection of the Body, and the Catholic view of the Antichrist.
The only work of the Apostolic Fathers associated with St. James to my knowledge is the Protoevangelium of St. James, supposedly written in 120 A.D. Of course, both of the Apostles named St. James were dead by then, but there is evidence that someone in St. James’ tradition wrote the book and was possibly one of his disciples – hence why his name appears in the title.
So that’s it for the Apostolic Fathers who we can trace to specific Apostles. Other Apostolic Fathers include the anonymous author of the Didache, Mathetes the author of the letter to Diognetus, and the anonymous author of Second Clement. It is worth noting that that work is sometimes associated with Pope St. Soter. He reigned about 100 years after most of the Apostles died, but it is barely possible that he had once heard the apostle John as a child, if he began his papacy when he was already in his late 70s (that scenario seems very unlikely). Second Clement is supposedly written by someone who really knew the Apostles, and Pope Soter’s connection to it, if there is one, may only mean that he had an older gentleman in his community who was an Apostolic Father, and he had him ghost-write the letter.
Regarding the author of the Didache and Mathetes, we do not know which Apostles they knew. But there is evidence that they did know some of them.
The Apostle St. Thomas went to India. The earliest Christian I know who is also associated with India is St. Clement of Aexandria, who traveled to India in the mid or late 100s to meet the Catholics there. When he returned to Alexandria he was influential in the early formation of a Catholic school known as the Catechetical School of Alexandria. He may have met some people in India who knew the Apostle.
I hope this helps. I wish I knew more so I could help more.