There are several other references to the Father, Son and Holy Spirit in the Bible. Luke 1:35, Mark 1:10-11, Hebrews 9:14 and Acts 7:55 all contain specific references to all three hypostases of God.
This loose understanding of the Trinity served the early Christians fairly well until the rise of the Arian heresy. The Arian movement rejected the Trinity, citing a belief that Jesus was created by God the Father, and thus not on an equal footing with him as Trinitarian belief requires. This would open the door to the first serious debates over the nature of the Trinity. It culminated in the dogmatic clarification of the nature of the Trinity, assuring the co-indwelling, co-inhering, and mutual interpenetration within the threefold nature of the Trinity. This status is referred to with the term perichoresis.
When Constantine the Great adopted Christianity as his religion (and therefore the religion of the Roman Empire) he demanded that there be a final ruling on the nature of God within the Christian Church. Various councils were held by the Church Fathers. The ultimate outcome was that the Church found the Trinitarian doctrine to be sound, and that the Father, Son and Holy Spirit were homoousios (meaning ‘of one essence’ – note the root of the word is ousia) as opposed to the Arian homoiousios (of similar essence).
This was not the last time this fragile and complex balance was to be irritated. If there is one thing that theologians do well, it is find different interpretations to the same rule. So while it is a relatively fixed teaching that God has one essence but three persons, the specifics of the interaction and roles of these three persons has been subject to debate.