The term "Trinity" does not appear in scripture. It is a term coined by theologians to attempt to keep the Jewish tradition of monotheism alive within Christianity.
The earliest form of Judaism was not monotheistic. The "Elohim" of the Old Testament was plural. Hence the English translation of "God" (in Hebrew "Elohim" a plural noun) saying "Let us make man in our image." To be true to the text it was necessary to employ a plural pronoun. Therefore, right at the beginning of the scriptural text God is plural.
Over time, the surrounding cultures with their excesses were competing with the religion of the Old Testament and Judaism developed a militant form of monotheism to reject the many gods of other religions.
But in the New Testament text it is apparent that Christ was here while another being, called by Christ "His Father in heaven" spoke from above. The Father's voice was heard at Christ's baptism and on the Mount of Transfiguration. The scriptures make them distinct beings.
Despite their separate existence, in John Chapter 17 Christ prays for all His followers and acknowledges that He (Christ) and the Father are "one." That "oneness" of the Father and Son has been used to claim there is but one God for Christians. However, the "oneness" of the Father and Son is explained by Christ in that same prayer. Christ asks that His disciples may be "one" just "as the Father and Son" are "one." Not by merging them into a single being, but by uniting them in faith and understanding. They become unified in belief, worship of God the Father and God the Son, but they remain separate and distinct persons.
The Trinity was an attempt to explain the separate personages of the "one true God" as one being, composed of one substance, and who is incomprehensible to the human mind. This unknowable and three-in-one being is described in one of the creeds adopted by the Catholics before the Protestant Reformation, and carried by the Protestants into ever one of the main Protestant churches. It states the following:
"That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. And yet they are not three eternals; but one eternal. As also there are not three uncreated; nor three infinites, but one uncreated; and one infinite. So likewise the Father is Almighty; the Son Almighty; and the Holy Ghost Almighty. And yet they are not three Almighties; but one Almighty. So the Father is God; the Son is God; and the Holy Ghost is God. And yet they are not three Gods; but one God."
THAT is the Trinity. It is unscriptural, not understandable, and makes distinct beings an exercise in word games.
God, on the other hand, is a Father and a Son who minister to mankind through the Holy Ghost, and they are separate, distinct personages. They are, however, "one" in their purpose, and in their harmonious effort to save mankind.