God
Christians inherited the Jewish view of God but, under the influence of Neoplatonism defined characteristics: aseity, immutability, omnipresence, impassibility, omniscience, omnipotence. Christians inherited the Platonic problem of "Participation"; how can the created interact with the Creator? The Logos stands between God and the world and The key was to link creation and redemption; the same Creator is the Redeemer, the one who spoke to Israel is the one who spoke in Jesus.
In worship, right from the beginning, Christians called Jesus "Lord", Kyrios, the word used in the Greek Septuagint for the Hebrew YHWH. Jesus was at the centre of worship, received the worship and obedience of the Church; people were baptised in the name of the Trinity. In spite of his Platonic formulation of God, Justin Martyr died for Christ.
Arius (250-336), said that the Logos was created within time but Nicea (325) said the Word incarnate in Christ was created before time began. Arius said that Jesus was not God but a creature, the first creature and the creator of all other creatures but still a creature. Many could not accept that they had suffered torture, and that their friends had died martyrs, for a creature. The key word was homousios - God and Jesus being "the same" - but some feared that the persons of the trinity might be reduced to 'faces' (modalism); but the imperial party supported Arianism because it did not like a crucified carpenter to be equal to God, let alone the Emperor! There was bitter controversy and persecution; a suggested compromise of "similar" or homoiousios failed; but if the Logos is a creature it can't be an intermediary between the mutable and immutable.
The critical point was worship practice where Christ was honoured equal to the Creator; this was a religious rather than a philosophical argument. The supporters of Athanasius of Alexandria (293-373) won but the argument rumbled on for another half century.
The broad agreement was that:
Synthesis: Trinity rejects polytheism; three can be addressed as God but not different from the one God; God is a mystery; the concept of oneness needs to be "corrected" so that it means at one in love not solo nor "above".
Matters might have settled for centuries but for the Filioque controversy. This word, meaning "and the son" was inserted into the Nicene Creed at the latter end of the 8th Century, possibly in Spain to assist with metrical chanting. The idea that the Spirit proceeds from both the Father and Son is Augustinian but the Greek speaking church tended to think of the Father as the source of the Spirit. The Formal schism between Greek and Latin church took place in 1054.
For reference, Augustine likened the Trinity to our memory, intellect and will.
"Process philosophers" like Alfred North Whitehead (1861-1947) developed the idea that the Trinity was not a fixed essence but a process.