Trinity of the Church Fathers
Trinitarian expressions which link together the names of the Father, Son, and Holy Spirit occurred very early in the History of the Christian Church. Two examples appear in the New Testament: 2 Corinthians 13:14 and Matthew 28:19. The context of 2 Corinthians 13:14 (verse 13 in the Vulgate), which is the close of a letter, suggests the church's conjunction of the Father, Son and Holy Spirit may have originated as a doxological formula; while the context of Matthew 28:19, the Great Commission, shows that the verbal conjunction of the Father, Son, and Holy Spirit was used early on as a baptismal formula.
This triadic pattern is even more marked in the glimpses available of the early Church's liturgy and day-to-day catechetical practice. The oldest extant work in which the word "Trinity" itself (Greek Trias, triados) is used to refer to Father, Son and Holy Spirit is Theophilus of Antioch's 2nd-century To Autolycus. The relationship between the Father, Son and Holy Spirit as one substance (ousia) and three co-equal persons (hypostaseis) was not formally ratified until the First Council of Constantinoplein 381.
Second century 
Early second century: Ignatius of Antioch 
Ignatius, second bishop of Antioch, who was martyred in Rome around 110 AD, wrote a series of letters to churches in Asia Minor on his way to be executed in Rome. The conjunction of Father, Son and Holy Spirit appears in his letter to the Magnesian church:
Study, therefore, to be established in the doctrines of the Lord and the apostles, that so all things, whatsoever ye do, may prosper both in the flesh and spirit; in faith and love; in the Son, and in the Father, and in the Spirit; in the beginning and in the end; with your most admirable bishop, and the well-compacted spiritual crown of your presbytery, and the deacons who are according to God. Be ye subject to the bishop, and to one another, as Jesus Christ to the Father, according to the flesh, and the apostles to Christ, and to the Father, and to the Spirit; that so there may be a union both fleshly and spiritual. —Epistle to the Magnesians, Chapter 13 [SR]
First half of second century or late first century: Didache 
This source uses the gospel of Matthew only and no other known gospel, and thus it must have been written before the four-gospel canon had become widespread in the churches, i.e. before the second half of the 2nd century when Tatian produced the Diatessaron. Given its literary dependence on the Gospel of Matthew, it is not surprising that the Didache follows the Gospel of Matthew in designating the Trinitarian formula as a baptismal formula:
After the foregoing instructions, baptize in the name of the Father, and of the Son, and of the Holy Spirit, in living [running] water…. If you have neither, pour water three times on the head, in the name of the Father, and of the Son, and of the Holy Spirit. —Didache 7:1
ca.151: Justin Martyr 
Even though Justin Martyr does not use the word "Trinity" explicitly, his First Apology, written around AD 150, reveals a primitive theology of the Trinity, in which God is in first place, Christ in second, and the Spirit in third:
We will prove that we worship him reasonably; for we have learned that he is the Son of the true God himself, that he holds a second place, and the Spirit of prophecy a third. For this they accuse us of madness, saying that we attribute to a crucified man a place second to the unchangeable and eternal God, the Creator of all things; but they are ignorant of the mystery which lies therein. —First Apology 13:5–6
Justin also wrote, "in the name of God, the Father and Lord of the universe, and of our Saviour Jesus Christ, and of the Holy Spirit." 
ca. 155: Polycarp of Smyrna 
Polycarp was martyred in Smyrna (where he was also Bishop) in the year 155. It is said by Irenaeus of Lyons that he was a pupil of the Apostle John. In his final prayer before his martyrdom, he "praises, glorifies, and blesses" the Father, Son, and Holy Spirit:
169-181: Theophilus of Antioch 
Theophilus of Antioch's Ad Autolycum is the oldest extant work that uses the actual word "Trinity" to refer to God, his Word and his Wisdom. The context is a discussion of the first three days of creation in Genesis 1-3:
It is the attribute of God, of the most high and almighty and of the living God, not only to be everywhere, but also to see and hear all; for he can in no way be contained in a place.... The three days before the luminaries were created are types of the Trinity, God, his Word, and his Wisdom. —To Autolycus 2:15
Third century: Trinitarian theology in response to Patripassianism and Sabellianism 
In the early 3rd century Tertullian and Hippolytus of Rome wrote Against Praxeas and Against Noetus, respectively, which may be considered the first extant expository treatments of Trinitarian theology. Both authors use the word Trinity (Latin: Trinitas; Greek: Trias). They wrote these works to combat Patripassianism, the view that the Father suffered on the cross along with the Son. In the 3rd century there were also Trinitarian theologies expressed in writings against Monarchianism, Sabellianism and Modalism.
216: Tertullian 
Tertullian's treatise against a Patripassian heretic named Praxeas, who claimed that the Father had suffered with the Son on the cross, is arguably the oldest extant treatise with a detailed explicit Trinitarian theology. In his Against Praxeas Tertullian wrote:
And at the same time the mystery of the oikonomia is safeguarded, for the unity is distributed in a Trinity. Placed in order, the three are the Father, Son, and Spirit. They are three, however, not in condition, but in degree; not in being, but in form; not in power, but in kind; of one being, however, and one condition and one power, because he is one God of whom degrees and forms and kinds are taken into account in the name of the Father, and of the Son, and of the Holy Spirit. —Against Praxeas 2
ca. 220: Hippolytus of Rome 
In the early 3rd century, Hippolytus of Rome wrote a treatise Against Noetus, in response to a Christian from Smyrna named Noetus who had been promoting Patripassian views, which Hippolytus deemed heretical. Noetus and other Patripassians, such as Praxeas (see above), claimed that the Father as well as the Son had suffered on the cross. Like Tertullian, Hippolytus explicitly used the word Trinity in his treatise against Patripassian views:
The Father's Word, therefore, knowing the economy and the will of the Father, to wit, that the Father seeks to be worshipped in none other way than this, gave this charge to the disciples after he rose from the dead: "Go ye and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost." (Matt 28:19) And by this he showed that whosoever omitted any one of these, failed in glorifying God perfectly. For it is through the Trinity that the Father is glorified. For the Father willed, the Son did and the Spirit manifested. —Against Noetus
ca. 225: Origen 
Origen's On First Principles (De Principiis or Peri Archon) is the oldest extant Christian theological treatise. Origen's Trinitarian theology is developed in this treatise, which reveals that by this time the use of the word Trinity to refer to Father, Son and Holy Spirit is standard in orthodox churches:
For it is the Trinity alone which exceeds every sense in which not only temporal but even eternal may be understood. It is all other things, indeed, which are outside the Trinity, which are to be measured by time and ages....It seems right to inquire into the reason why he who is 'born again through God' to salvation has need of both Father and Son and Holy Spirit and will not obtain salvation apart from the entire Trinity, and why it is impossible to become partaker of the Father or the Son without the Holy Spirit. In discussing these points it will undoubtedly be necessary to describe the activity which is peculiar to the Holy Spirit and that which is peculiar to the Father and Son. —
ca. 256: Novatian 
Novatian, presbyter of Rome, wrote the oldest extant Christian treatise that is specifically dedicated to and entitled On the Trinity. It was written in response to a number of views deemed heretical by Novatian, and particularly against Sabellius, who had maintained that the Trinity was divided into three prosopa, or "characters by which God is revealed to man, the Trinity being one of revelation, not essence".
For Scripture as much announces Christ as also God, as it announces God himself as man. It has as much described Jesus Christ to be man, as moreover it has also described Christ the Lord to be God. Because it does not set forth him to be the Son of God only, but also the son of man; nor does it only say, the son of man, but it has also been accustomed to speak of him as the Son of God. So that being of both, he is both, lest if he should be one only, he could not be the other. For as nature itself has prescribed that he must be believed to be a man who is of man, so the same nature prescribes also that he must be believed to be God who is of God…. Let them, therefore, who read that Jesus Christ the son of man is man, read also that this same Jesus is called also God and the Son of God. —Treatise on the Trinity, 11
262: Pope Dionysius 
According to Athanasius of Alexandria, in the mid-3rd century Pope Dionysius wrote a letter to Dionysius of Alexandria criticizing Sabellius's views on the relations between the Son and the Father, as well as some who attempted to refute Sabellius's views. He quotes parts of Dionysius' letter in On the decrees of the Council of Nicaea . In this letter it is clear that Dionysius used the word Trinity (Greek Trias) to explicate the relations between Father, Son and Holy Spirit:
Next, I may reasonably turn to those who divide and cut to pieces and destroy that most sacred doctrine of the Church of God, the Divine Monarchy, making it as it were three powers and partive subsistences and godheads. I am told that some among you who are catechists and teachers of the Divine Word, take the lead in this tenet, who are diametrically opposed, so to speak, to Sabellius' opininons; for he blasphemously says that the Son is the Father, and Father the Son, but they in some sort preach three Gods, as dividing the sacred Unity into three subsistences foreign to each other and utterly separate. For it must be that with the God of the Universe, the Divine Word is united, and the Holy Ghost must repose and habitate in God; thus in one as in a summit, I mean the God of the Universe, must the Divine Trinity be gathered up and brought together....Neither, then, may we divide into three godheads the wonderful and divine Unity...Rather, we must believe in God, the Father Almighty; and in Christ Jesus, his Son; and in the Holy Spirit; and that the Word is united to the God of the universe. 'For,' he says, 'The Father and I are one,' and 'I am in the Father, and the Father in me'. For thus both the Divine Trinity and the holy preaching of the Monarchy will be preserved. —'De decretis Nic. syn.26
265: Gregory the Wonderworker 
There is one God.... There is a perfect Trinity, in glory and eternity and sovereignty, neither divided nor estranged. Wherefore there is nothing either created or in servitude in the Trinity; nor anything super-induced, as if at some former period it was non-existent, and at some later period it was introduced. And thus neither was the Son ever wanting to the Father, nor the Spirit to the Son; but without variation and without change, the same Trinity abides ever. —Declaration of Faith.
- Kelly, J.N.D. Early Christian Doctrines A & C Black 1965 (1965) p.88
- Eusebius of Caesarea, Church History iii.36
- St. Ignatius of Antioch to the Magnesians (Shorter Recension), Roberts-Donaldson translation.
- Date according to Metzger, Bruce. The Canon of the New Testament. 1997
- First Apology
- First Apology, 61 LXI
- To Autolycus
- Against Praxeas
- Against NoetusCh. 1
- Against Noetus Ch. 14.
- De Principiis, book 1, chapter 3
The Trinity according to Origen
- Jerome, On Illustrious Men Ch. 70
- J.E. Oulton, Eusebius: Ecclesiastical History Vol 2; (Cambridge, 1980 reprint; p. 143, n. 1).
- Fathers/Volume V/Novatian/A Treatise of Novatian Concerning the Trinity/Part 11|Treatise on the Trinity, part 11
- Johannes Quasten, Patrology, Vol. 2, Utrecht, 1964, p. 239-241.
- Athanasius De decretis Nic. syn. 26.
- Jerome, On Illustrious Men Ch. 65
- Basil of Caesarea discusses the occasion for Gregory writing this Declaration of Faith in Letter 205
- Declaration of Faith