1) Could you begin to demonstrate that the Bible actually teaches the Trinity even though it never uses the word
The Trinity and doctrine of Two Natures are consequences of the very scriptural Incarnation of Jesus Christ - God the Son.
Without a doubt the central affirmation of the Christian faith is that Jesus Christ is both divine and human. On this truth all Christians are agreed, since indeed one who denies this can scarcely bear the name Christian in any meaningful sense at all.
Fairbairn, Donald Grace and Christology in the Early Church [Oxford Early Christian Studies, 2003 p.vii]
And the Word became flesh and dwelt among us, and we beheld his glory, glory as of the Only-Begotten of the Father, full of grace and truth.
John 1:14
Although he existed in the form of God, he did not consider equality with God something to be grasped, but he emptied himself by taking the form of a servant and being made in human likeness.
Phil. 2:6-7
These two biblical texts proclaim what is at once both the central truth of the Christian faith and perhaps its most controversial affirmation. God has become a man. The one who was equal with God has taken upon himself the form of a servant. In the man Jesus who has lived among us, we see the glory of the Fathers only Son. As early as ad 325, this affirmation was enshrined prominently in the Creed of Nicea, even though the Council of Nicea dealt not with the incantation per se, but with the full deity of God the Son. The Creed affirms not only that we believe in one God, Father, all-sovereign, maker of all things seen and unseen; and in one Lord Jesus Christ, the Son of God but also that this Son of God for us men, and for our salvation, came down, and was incarnated, and was made man, suffered, and arose on the third day.
Fairbairn, Donald Grace and Christology in the Early Church [Oxford Early Christian Studies, 2003 p.1]
2) Could you provide an Old Testament defence of this? Why is their no awareness of this idea before Christ in the Jewish community?
See Secundulus' post above.
3) What does the Trinity say about the nature of God- what insight does it give us into God?
That God loved us such that he Incarnated, suffered and took on sin for our sake, undeserving of His salvific Grace as we are.
Thus, prior to the Nestorian controversy, the church had opposed three extreme misrepresentations of the person and work of Christ. (1) Christ was a divine being and therefore could not suffer (Docetism); (2) God the Father was temporarily changed into the suffering Son, at the expense of his full divinity and transcendence (Patripassianism); (3) Christ was involved in change, birth, suffering, and death, therefore he could not be fully divine (Arianism). Having ruled out the three extreme options, the church asserted that the Son of God suffered in reality and not mere appearance; that it was the Son who became incarnate and suffered, not the Father; that the Sons involvement in suffering did not diminish his divine status, because the incarnation was a supreme act of divine compassion and as such it was most appropriate and God-befitting.
The justification of the incarnation as an act worthy of God is a common theme of Christian apologetic against philosophically minded pagans, whose understanding of God did not allow for the possibility that God could empty himself, assume the human condition, and suffer the consequences. The very fact that the Fathers quite self-consciously understood their argument for the God-befitting character of the incarnation to be directed against Hellenistic philosophers puts into question the assumption that the Fathers asserted divine impassibility simply as a result of their uncritical acceptance of the conceptuality of Hellenistic theological thought.
Gavrilyuk, Paul L. The Suffering of the Impassible God The Dialectics of Patristic Thought [Oxford Early Christian Studies, 2004 p. 18]
4) What does it mean for our Christian lives to believe in a Trinitarian God as opposed to a Monotheistic one like the Jews or Muslims believe in? How does believing in the Trinity distinguish us from Polytheists also?
A solitary God would not be Love without limits. A God who made himself twofold, according to a pattern common in mythology, would make himself the root of an evil multiplicity to which he could only put a stop by re-absorbing it into himself. The Three-in-One denotes the perfection of Unity of Super-unity, according to Dionysius the Areopagite fulfilling itself in communion and becoming the source and foundation of all communion. It suggests the perpetual surmounting of contradiction, and of solitude as well, in the bosom of an infinite unity.
Clément, Olivier The Roots of Christian Mysticism [NCP 1982, 1993 pp74-75]
When I say God, it is of the Father, of the Son and of the Holy Spirit that I wish to speak without diluting the godhead beyond these limits, lest I should introduce a whole tribe of divinities, and without restricting it to something less than the three persons, lest I be accused of impoverishing the godhead. Otherwise I should fall into the simplicity of the Judaizers or into the multiplicity of the Hellenizers . . .
Thus the Holy of Holies, enveloped and veiled by the Seraphim, is glorified by a threefold consecration in the unity of the godhead.
Gregory Nazianzen Oration 54, For Easter, 4 (PG 36, 620) quoted from Clément, Olivier The Roots of Christian Mysticism [1982, 1993 p66]
5) With what simple analogies would you try and explain the Trinity to someone else?
Simple analogies for the infinite mystery beyond all human understanding? Beyond my capacity, I'm afraid.
The One enters into movement because of his fullness.
The Two is transcended because the godhead is beyond all opposition.
Perfection is achieved in the Three who is the first to overcome the compositeness of the Two.
Thus the godhead does not remained confined, nor does it spread out indefinitely.
Gregory Nazianzen Oration 23,8 (PG 35, 1160)
Gregory explains the Trinity by means of a double contrast. He contrasts it on the one hand with the ancient pagan notion of diffused divinity, and on the other with the God of Judaism who is totally transcendent and distinct (although this statement would certainly need qualification in the light of the mystical tradition of Judaism). Today, from the same standpoint, we could contrast the mystery of the trinity with, on the one hand, the exoteric rigid monotheism of popular Judaism and Islam, and, on the other, with Hinduism and Buddhism and their transpersonal concept of divinity wherein everything is engulfed, as in an immeasurable womb which is why India loves to speak of the divine Mother.
Here too some qualification would be necessary in speaking about both the way of love in India and the various Buddhist interpretations of Grace. However, the mystery of personality and inner life, a synthesis of which is sought with difficulty by the various religions, seems to have been resolved in the doctrine of the Three-in-One, where association is perceived within the absolute, or rather the absolute within the association.
We are made in the image of God. From all eternity there is present in God a unique mode of existence, which is at the same time Unity and the Person in communion; and we are called to realize this unity in Christ, when we meet him, under the divided flames of the Spirit. Therefore we express the metaphysics of the person in the language of Trinitarian theology. What could be called the Trinitarian person is not the isolated individual of Western society (whose implicit philosophy regards human beings as similar but not consubstantial). Nor is it the absorbed and amalgamated human beings of totalitarian society, or of systematized oriental mysticism, or of the sects. It is, and must be, a person in a relationship, in communion. The transition from divine communion to human communion is accomplished in Christ who is consubstantial with the Father and the Spirit in his divinity and consubstantial with us in his humanity.
Clément, Olivier The Roots of Christian Mysticism [1982, 1993 p65-66]