1) The doctrine of the Trinity is the logical result of the the New Testament witness about God. As Tom Wright says somewhere, if the Trinity hadn't existed, the church would have had to invent it. The Trinity is the only way to make sense of what the New Testament writers claim about Jesus (the Son) and the Holy Spirit, and the relationship of the Son to the Father.
2) The doctrine of the Trinity has practical implications for the doctrine of salvation. Contrary to some who have suggested that Trinitarian speculation in the early church was a distraction and somehow beside the point, the early fathers and mothers in the faith understood that at the heart of the Trinitarian controversies nothing less than the salvation of humanity was at stake. At the heart of it all was this question: "If we say thus and so about God, what does it means for our salvation?" To affirm that Jesus is fully divine and fully human is to rightly understand that only God can save and that God cannot save that which God has not become. It is no accident that one of the earliest proclamations of the first Christians was that Jesus is Savior.
3) The early Christians also proclaimed that Jesus is Lord. Such a proclamation is steeped in the Old Testament context of God's lordship over all the world. It is not a coincidence that the Apostle Paul takes Old Testament Scriptures that clearly refer to Yahweh and applies them to Jesus (e.g. Philippians 2:6-11).
4) If one studies the history of the early church councils it appears that the discussion of the Holy Spirit comes into the picture almost incidentally. There is some truth to that observation. Nevertheless, there is good reason that the Spirit is by necessity included in the Trinitarian formulations. God the Father and the Son are present in the Spirit's workings in the world. In John's Gospel Jesus tells the disciples that he must go away if the Spirit is to come (16:7), but before this Jesus informs them that with the coming of the Spirit, he will come to them (John 14:18).
5) The doctrine of the Trinity affirms that God is by nature relational. This explains why the universe, including human beings, are intrinsically relational. Theoretical physicist turned Anglican priest and theologian, John Polkinghorne writes in his book, Science and the Trinity:
With physics moving in a more holistic direction, we might expect by analogy the need to challenge the individualistic atomism that is so characteristic of contemporary thinking about human nature. If electrons are counterintuitively entangled with each other, we may need to contemplate the possibility that persons participate in some greater solidarity than atomised Western society is able to recognize. Such an insight is surely consonant with the Christian understanding of the community of the faithful as the Body of Christ, constituting a web of relationality vastly more comprehensive than the one-to-one exchange of I and Thou.
These remarkable developments in relational and holistic thinking that are taking place within the fold of science are deeply congenial to Trinitarian ways of thought.. They by no means "prove" the Trinity, but they are profoundly consonant with a theology of nature that sees the relation of perichoretic exchange between the divine Persons as lying at the heart of the Source of all created reality. One could paraphrase the title of John Zizioulas's insightful book on Trinitarian theology Being as Communion, by the phrase, "Reality is relational" (75).Thus, even the universe itself may be a vestigia trinitatis.
There is indeed good reason to affirm that the Trinity IS the Christian doctrine of God. That is why Trinity Sunday is important. More can be said, but this is sufficient.
In Conclusion : To neglect the doctrine of the Trinity is to neglect the essence of our Christian faith in the God who created all that is and who has stamped God's image into the universe and even more clearly in humanity, and most clearly and decisively in Jesus Christ, who is the image of the invisible God-- God with a human face.