Questions on the Trinity for Muslims - Who were some of the early followers of Jesus that taught the doctrines of the Trinity?
Here are 16 writers I found, plus three others that mention the Father, Son, and Holy Spirit.
Ignatius, a disciple of the apostle John, frequently referred to Jesus as God. For example, in the Second Letter of Ignatius to the Ephesians chapter 9 Ignatius says, "And ye are prepared for the building of God the Father, and ye are raised up on high by the instrument of Jesus Christ, which is the cross; and ye are drawn by the rope, which is the Holy Spirit; and your pulley is your faith, and your love is the way which leads up on high to God" (Ante-Nicene Fathers volume 1 p.101)
Letter to Diognetus(c.130 A.D.) written by a a disciple of the apostles chapter 7 wrote "As a king sends his son, who is also a king, so sent He Him; as God He sent Him; as to men He sent Him; as a Saviour He sent Him."
Irenaeus (120-202 A.D.) "But that He [Jesus] is Himself in His own right, beyond all men who ever lived, God, and Lord, and King Eternal, and the Incarnate Word, proclaimed by all the prophets, the apostles, and by the Spirit Himself, may be seen by all who have obtained to even a small portion of the truth." (Irenaeus Against Heresies 3:19:2).
"Know thou that every man is either empty or full. For if he has not the Holy Spirit, he has no knowledge of the Creator; he has not received Jesus Christ the life; he knows not the Father who is in heaven;..." (Against Heresies 3:16)
"She [the church] also believes these points [of doctrine] just as if she had but one soul.... For the churches which have been planted in Germany do not believe or hand down anything different nor do those in Spain nor those in Gaul, nor those in the East nor those in Egypt nor those in Libya, nor ..."
Justin Martyr (wrote about 135-165 A.D) in his Dialogue with Trypho the Jew chapters 57-65, also gives a very good explanation of the Trinity, and why we should worship Jesus, though without actually using the word Trinity.
Bishop Theophilus of Antioch, (168-181/188 A.D.) mentioned the Trinity in his Letter to Autolycus 2:15.
Tertullian (200-220 A.D.) wrote extensively about the Trinity, in his letter Against Praxeus.
Clement of Alexandria (wrote 193-217/220 A.D.) also spoke of "the Holy Trinity" in Stromata 5:14.
Novatian (210-280 A.D.) from Rome wrote 32 chapters in Treatise on the Trinity.
Hippolytus (225-235/6 A.D.) after quoting part of John 1:1
"If, then the Word was with God and was also God what follows? Would one say that he speaks of two Gods? I shall not indeed speak of two Gods but of one; of two Persons however and of a third economy (disposition), viz., the grace of the Holy Ghost. For the Father indeed is One but there are two Persons because there is also the Son; and then there is the third the Holy Spirit. The Father decrees, the Word executes and the Son is manifested, through whom the Father is believed on. The economy of the harmony is led back to one God; for God is One. It is the Father who commands and the Son who obeys and the Holy Spirit who gives understanding; the Father is above all, and the Son who is through all and the Holy Spirit who is in all. And we cannot otherwise think of one God, but by believing in truth in Father and Son and Holy Spirit." Against the Heresy of One Noetus chapter 14.
Origen (230-254 A.D.) mentions the Trinity in de Principiis book 1 3:7.
Gregory Thaumaturgus (240-265 A.D.) in A Declaration of Faith mentions the Father, Son, and he mentions the Trinity three times in his "Declaration of Faith" in the Ante-Nicene Fathers vol.6 p.7
Dionysius of Alexandria (246-265 A.D.) mentions the Trinity by name twice in Letter 4 ch.8 p.93.
Bishop Munnulus of Girba uses the word "Trinity and quotes Matthew 28:19 "...in the name of the Father, of the Son, and of the Holy Spirit" The Seventh Council of Carthage (258 A.D.) p.567
Bishop Euchratius of Thenae uses the word Trinity and quotes Matthew 28:19 "...in the name of the Father, of the Son, and of the Holy Spirit" The Seventh Council of Carthage (258 A.D.) p.568
Dionysius of Rome (259-269 A.D.) says that the Trinity is declared in Scripture, but three gods is neither taught in the Old or New Testament. Against the Sabellians ch.1 in Ante-Nicene Fathers vol.7 p.365. He also mentions the Trinity in ch.3 p.366.
Thus those who followed the teachings of Jesus were very outspoken in affirming the Trinity of the Father, Son, and Holy Spirit.
The Trinity, like faith, goes beyond what people could have reasoned out for themselves in their own minds, but the Trinity does not go against reason and logic. As the preceding shows, many concepts of the Trinity have approximate parallels in Islam.