Where did the notion of a trinity begin?
Did the understudies of the Apostles mention the trinity?
Polycarp, Clement, and Ignatius were the students of the original apostles. They lived at the turn of the century, before and after 100 AD. Did they mention a trinity or give a description of a trinity in all their writings? No. Did they mention the Holy Spirit being a ground of consciousness distinct from the Father or Son? No. Did they mention the Holy Spirit being a person? No.
What about the next generation of writers, the Apologists? The Apologists wanted to demonstrate the Christian's faith as being supposedly compatible with Greek philosophy, ignoring the warnings of Paul (Colossians 2:8). In this era which included the later part of the 100's AD was Justin Martyr, Theophilus of Antioch, Tatian, and Athenagoras. Totally missing from their writings was any mention of a trinity. There was one bishop however, Irenaeus, of the same period of time, who mentioned "His Wisdom" as being a divine entity. He did this in reaction to gnosticism.
It was not until the 200's AD that even the primitive idea of a trinity began to take shape.
Quintus Septimius Florens Tertullianus, known as Tertullian, in the early 200's AD was the first to mention the primitive form of a trinity. He is the first person to use the word "trinity" and he was the first person to formulate the idea of one substance having three persons. He had the idea of the Spirit being a persona, not in the sense of a person but in the sense of a mask, which ancient actors would wear in a live play. Tertullian considered the Holy Spirit to be the "deputy" of the Son.
The idea of individual subsistence, hypostasis (from the Greek), was first introduced by Origen. Origen considered the Son to be derivative of the Father and the Spirit not to be co-equal to the Father or the Son. This idea (of the Son being derivative of the Father) would be latched onto by Arius in the 300's AD to eventually cause a major crisis leading to the Counsel of Nicea.
In the 300's AD, the Roman Emperor Constantine "converted" to Christianity. He made Christianity (the large group of what was called Christianity at that time) the official religion of Rome. By this time the religion (Christianity) had no resemblance to the church in which the Apostles were involved.
Arius in the 300's AD gained many followers as he taught that Christ was a created being, created by the Father. This was the birth of the idea of the unity (the idea of one God (the Father) denying the divinity of Christ). Arius made such an impact he created a crisis that led the Roman Emperor Constantine, at the behest of Pope Dionysius, to call together the Counsel of Nicea.
Up to that point in time, Monarchianism with monotheistic imperatives was pitted against those who believed the God head to be composed of more than one. There was the binity (God is a Kingdom consisting of 2 beings, Father and Son). There was the divine trias, which was the predecessor to the belief in the trinity. And most of all, there were the Arians (the followers of Arius).
So the Counsel of Nicea was called together in 325 AD. About 300 Catholic Bishops were brought together to resolve the dispute. The conclusion was the repudiation of Arianism.
The creed that came out of the Counsel of Nicea in 325 AD did not explicate the trinity. It simply proclaimed the divinity of Christ, rejecting Arianism. There was no resolution on the "personhood" of the Holy Spirit. That notion would not arise in full strength until the Counsel of Constantinople in 381 AD.
Basilius, commonly known as Basil, bishop of Caesarea in the later 300's AD, formulated ideas as to what the Holy Spirit was. This was mainly in reaction to Arius who was his enemy doctrinally. Basil and others such as Gregory of Nazianzus and Gregory of Nyssa were encouraged to develop ideas to combat the idea of arianism. The person who encouraged them was Athanasius who hated Arianism and wanted to wipe it out.
Gregory of Nazianzus acknowledged that the scripture does not refer to the Holy Spirit as God. Therefore they (he and the other Gregory and Basil) would have to pursue a line of logic outside of scripture in order to formulate the Holy Spirit to be more than it really is.
Using human reasoning, Basil concluded that spirit itself necessarily meant unchangeable. This is contrary to the teaching of scripture (some angels, for example, changed -- they followed Satan!). Nevertheless it appealed to the academic minds of that day. Also using human reasoning, Basil concluded that if the Spirit was sanctified, how could it be anything less than divine. The next logical step (also using human reasoning) was to declare the Holy Spirit a person. And that is exactly what he did.
This reasoning was the basis for the creed that came out of the Counsel of Constantinople in 381 AD. This creed declared the Holy Spirit to be co-equal and co-essential to God the Father and God the Son. It was the origin of the doctrine of God in three persons, holy trinity, the doctrine that is popular even in today's information age.