The Apostles Creed, referenced as early as 390 A.D., quite clearly articulates an understanding of our faith revolving around the three persons we identify as the members of the Trinity as core to the faith. It makes up almost half of the total text.
I believe in God, the Father Almighty,
the Maker of heaven and earth,
and in Jesus Christ, His only Son, our Lord:
Who was conceived by the Holy Ghost,
born of the virgin Mary,
suffered under Pontius Pilate,
was crucified, dead, and buried;
It is clearly understood both today and in its time that this creed was affirming a belief in One God and at the same time identifying how he is revealed to us. While it doesn't use the term Trinity nor define the mixed natures as clearly as later creeds and confessions, it is unlikely that such a creed would have seen the near universal acceptance as articulating the core of Christian belief if it went against the teachings of the church fathers of the previous three centuries. The lack of controversy on this topic is remarkable. We do know people who taught otherwise, such as the Gnostics and Arians, but part of the purpose of this Creed was to emphatically reject them, a standard that has continued throughout mainstream Christian history.
Before moving on, it would be remiss not to point out that the most emphatic and authoritative writings on this matter in the first couple centuries are in fact the Scriptures themselves. Looking in material written by "early fathers" for clues about what they did or did not believe without first and foremost considering the teachings found in the pages of the Bible their actions as church fathers helped to canonize would be a disservice to them.
That being said we do have clear evidence that the early Fathers did believe in the Trinity. Few of them used that word, but the words they did use fit the formula. For example Polycarp lived from 70 A.D to somewhere in the mid second century. Here's how he prayed:
"O Lord God almighty... I bless you and glorify you through the eternal and heavenly high priest Jesus Christ, your beloved Son, through whom be glory to you, with Him and the Holy Spirit, both now and forever."
While that statement was simply an affirmation of the three persons without directly trying to define their relation to each other, other early Fathers such as Theophilus from the late second century did remark on how they relate:
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.
Likewise Justin (another Martyr and church leader from the second century) had this to say about baptism:
"For, in the name of God, the Father and Lord of the universe, and of our Savior Jesus Christ, and of the Holy Spirit, they then receive the washing with water."
Clearly the three persons were at the center of the faith people were being inaugurated into. In another place, he comments specifically on the relation of these three persons to each other:
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.
By the first decade of the third century, we have quite a collection of declarations about the Trinity. Here is Tertullian:
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.
The list goes on. I can't quote them all here but Ignatius the Bishop of Antioch, Irenaeus, a disciple of Polycarp and later Bishop of Lyons, Didache, Hippolytus, Novatian, Origen, and other Fathers through the second and third centuries can all be quoted with emphatic articulations of the Trinity.
The exact formulation and best way to articulate this mystery has been widely debated in every age and language since. The above Fathers certainly had varying ideas about just what it meant to have a single God with three persons. We still struggle to wrap our heads around this today. And that's ok as long as we do not err on the side of either denying either the unity or diversity of his nature. We must not make out the persons of the Trinity to be separate beings nor deny that the one being is revealed to us in thee persons.