The Trinity in the Old Testament

There's no explicit mention of the Trinity in the Old Testament (obviously, otherwise the Jews would believe it as well). However, there are enough 'glimpses' of the teaching that we can say that it is clearly being foreshadowed. Here follow the main glimpses of the Trinity doctrine.

God in the plural

Genesis 1,1 states: In the beginning God created the heaven and the earth. The word used for God, Elohim, is a plural noun but used here with a singular verb bara.

Then Genesis 1,26: And God said, Let us make man in our image, after our likeness. God speaks to Himself in the plural.

Genesis 19,24 Then the Lord rained on Sodom and Gomorrah sulfur and fire from the Lord out of heaven. The Lord that acts is distinguished from the Lord of origin.

Genesis 35,1 God said to Jacob, Arise, go up to Bethel, and settle there. Make an altar there to the God who appeared to you when you fled from your brother Esau. The God that speaks is distinguished from the One who appeared before.

And Isaiah 54,5 proclaims: For thy Maker is thine husband; the Lord of hosts is his name; and thy Redeemer the Holy One of Israel; The God of the whole earth shall he be called. The word used for Maker is plural in Hebrew, and there's a hint that the Maker and the Redeemer are different persons.

Jeremiah 50,40 As when God overthrew Sodom and Gomorrah and their neighbors, says the Lord, so no one shall live there, nor shall anyone settle in her. The One speaking is distinguished from Who acted.

Triplets

In Genesis 18,3 Abraham is visited by three men which he adresses with Lord and in the singular: And said, My Lord, if now I have found favour in thy sight, pass not away, I pray thee, from thy servant.

In Genesis 18,10 the response is in the singular as well: And he said, I will certainly return unto thee according to the time of life; and, lo, Sarah thy wife shall have a son.

Three lines later, in Genesis 18:13, it is made clear that it is really God who is speaking: And the Lord said unto Abraham, Wherefore did Sarah laugh,...

When in Genesis 48,15-16 Jacob blesses his son Joseph, he uses the name of God three times and each time identified differently: And he blessed Joseph, and said, God, before whom my fathers Abraham and Isaac did walk, the God which fed me all my life long unto this day, The Angel which redeemed me from all evil, bless the lads;

In the subsequent chapter, Genesis 49,23-25, a similar triple blessing is give: But [Joseph's] bow abode in strength, and the arms of his hands were made strong by the hands of the mighty God of Jacob; (from thence is the shepherd, the stone of Israel:) Even by the God of thy father, who shall help thee; and by the Almighty, who shall bless thee with blessings of heaven above, blessings of the deep that lieth under, blessings of the breasts, and of the womb.

Numbers 6,24-27 gives a triple benediction as well: The Lord bless you and keep you; the Lord make his face shine upon you and be gracious to you; the Lord turn his face toward you and give you peace. So they will put my name on the Israelites, and I will bless them.

Deuteronomy 6,4 contains the great Sh’ma prayer of Judaism: Hear, O Israel: The Lord our God is one Lord. Besides a triple naming (Lord, God, Lord), the word used for "one" signifies unity not singleness and so is also a hint of the Trinity.

Isaiah 6,3 also gives a hint of the Trinity: And they were calling to one another: Holy, holy, holy is the Lord Almighty; the whole earth is full of his glory. 

Finally, in Jeremiah 7,2-4 the message that the prophet is to proclaim, first contains Lord three times, and then "temple of the Lord" another three times: Stand in the gate of the Lord's house, and proclaim there this word, and say, Hear the word of the Lord, all ye of Judah, that enter in at these gates to worship the Lord. Thus saith the Lord of hosts, the God of Israel, Amend your ways and your doings, and I will cause you to dwell in this place. Trust ye not in lying words, saying, The temple of the Lord, The temple of the Lord, The temple of the Lord, are these.

Angel of the Lord  

Already the blessing of Joseph above uses this title, the revelation of a person distinct from God, who is nonetheless equal to God. Other examples: 

Genesis 16,7+13 The angel of the Lord found her by a spring of water in the wilderness + So she named the Lord who spoke to her, You are El-roi for she said, Have I really seen God and remained alive after seeing him? 

Genesis 31,11+13: And the angel of God spake unto me in a dream + for I have seen all that Laban doeth unto thee. I am the God of Bethel.

Again in Exodus 3,2+4: And the angel of the Lord appeared unto him in a flame of fire out of the midst of a bush + And when the Lord saw that he turned aside to see, God called unto him out of the midst of the bush, and said, Moses, Moses.

In Exodus 23,20-21 the angel of the Lord has His Name and the divine power to forgive sins: See, I am sending an angel ahead of you to guard you along the way and to bring you to the place I have prepared. Pay attention to him and listen to what he says. Do not rebel against him; he will not forgive your rebellion, since my Name is in him.

Joshua 5,15 The commander of the army of the Lord said to Joshua, Remove the sandals from your feet, for the place where you stand is holy. The same words that God spoke to Moses from the burning bush.

Judges 2,1 Now the angel of the Lord went up from Gilgal to Bochim, and said, I brought you up from Egypt, and brought you into the land that I had promised to your ancestors. I said, I will never break my covenant with you. The angel claims Gods covenant as his own.

And Judges 6,22-23 Then Gideon perceived that it was the angel of the Lord; and Gideon said, Help me, Lord God! For I have seen the angel of the Lord face to face. But the Lord said to him, Peace be to you; do not fear, you shall not die.

Again in Judges 13,21-22 The angel of the Lord did not appear again to Manoah and his wife. Then Manoah realized that it was the angel of the Lord. And Manoah said to his wife, We shall surely die, for we have seen God.

Hosea 12,4-5 He struggled with the angel and overcame him; he wept and begged for his favor.

He found him at Bethel and talked with him there—the Lord God Almighty, the Lord is his name! The angel is identified as God Himself.

Zechariah 1,8-11 During the night I had a vision, and there before me was a man mounted on a red horse. He was standing among the myrtle trees in a ravine. Behind him were red, brown and white horses. I asked, What are these, my lord? The angel who was talking with me answered, I will show you what they are. Then the man standing among the myrtle trees explained, They are the ones the Lord has sent to go throughout the earth. And they reported to the angel of the Lord who was standing among the myrtle trees. The angel is also a man and with authority as the men God sent out report to him.

Finally, in Zechariah 3,4 the angel has the divine power to forgive sins: The angel said to those who were standing before him, Take off his filthy clothes. Then he said to Joshua, See, I have taken away your sin, and I will put fine garments on you. 

Holy Spirit

There are multiple mentions of "God's Spirit" in the Old Testament referring to a separate person, the first already in its second verse (Genesis 1,2): And the Spirit of God moved upon the face of the waters.

A few chapters onward, in Genesis 6,3, it says: Then the Lord said, "My Spirit shall not abide in mortals forever, for they are flesh; their days shall be one hundred twenty years." 

In the next book, Exodus 31,3, the Lord speaks to Moses: And I have filled him with the Spirit of God, in wisdom, and in understanding, and in knowledge, and in all manner of workmanship.

The prophets also mention the Spirit of the Lord, e.g. Zechariah 4,6: Then he answered and spake unto me, saying, This is the word of the Lord unto Zerubbabel, saying, Not by might, nor by power, but by my Spirit, saith the Lord of hosts.

The greatest Jewish prophet mentions Him even multiple times, the first time is in the Messianic prophecy of the "stump of Jesse" (Isaiah 11,2-3): The Spirit of the Lord will rest on him - the Spirit of wisdom and of understanding, the Spirit of counsel and of power, the Spirit of knowledge and of the fear of the Lord - and he will delight in the fear of the Lord. 

The next one is Isaiah 48,16: Come ye near unto me, hear ye this; I have not spoken in secret from the beginning; from the time that it was, there am I: and now the Lord God, and his Spirit, hath sent me.

And the prophecy that Jesus Himself quotes in the synagogue of Nazareth (Isaiah 61,1): The Spirit of the Lord God is upon me; because the Lord hath anointed me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound;

Finally, Isaiah 63,10 states: Yet they rebelled and grieved his Holy Spirit. So he turned and became their enemy and he himself fought against them. 

Shekhinah (Glory/Cloud) of God 

The Shekhinah is another prefiguration of the Holy Spirit, a female word signifying God's dwelling in Exodus 40,34 when the Tabernacle was ready: Then a cloud covered the tent of the congregation, and the Glory of the Lord filled the tabernacle.

And later it was present in the Temple, 1 Kings 8,10 reads: And it came to pass, when the priests were come out of the holy place, that the Cloud filled the house of the Lord.

Cf. Luke 1:35 where the Holy Spirit, the Power of God overshadows the virgin Mary.

Lady Wisdom

Another prefiguration of the Holy Spirit: Lady Wisdom was present before creation (Proverbs 8,22-30): The Lord possessed me in the beginning of his way, before his works of old. I was set up from everlasting, from the beginning, or ever the earth was. When there were no depths, I was brought forth; when there were no fountains abounding with water. Before the mountains were settled, before the hills was I brought forth: While as yet he had not made the earth, nor the fields, nor the highest part of the dust of the world. When he prepared the heavens, I was there: when he set a compass upon the face of the depth: When he established the clouds above: when he strengthened the fountains of the deep: When he gave to the sea his decree, that the waters should not pass his commandment: when he appointed the foundations of the earth: Then I was by him, as one brought up with him: and I was daily his delight, rejoicing always before him;

And is the Breath and Power of God (Wisdom 7,24-26): For wisdom is more moving than any motion: she passeth and goeth through all things by reason of her pureness. For she is the breath of the power of God, and a pure influence flowing from the glory of the Almighty: therefore can no defiled thing fall into her. For she is the brightness of the everlasting light, the unspotted mirror of the power of God, and the image of his goodness. 

Son of God/the Word/Messiah 

Multiple prefigurations of Jesus as the Son of God, as the incarnate Word (cf. John 1,1) and as divine Messiah are equally present in the Old Testament, such as God walking in the garden of Eden (Genesis 3,8): Then the man and his wife heard the sound of the Lord God as he was walking in the garden in the cool of the day.

Genesis 15,1 After these things the Word of the Lord came to Abram in a vision, Do not be afraid, Abram, I am your shield; your reward shall be very great. The Word itself speaks.

1 Samuel 3,21 The Lord continued to appear at Shiloh, for the Lord revealed himself to Samuel at Shiloh by the Word of the Lord. The Lord and his Word are distinguished.

1 Kings 19,9 Then the Word of the Lord came to him, saying, What are you doing here, Elijah?  Again it's the Word itself that speaks.

Some are by King David (Psalm 2,6-7): I have set my king on Zion, my holy hill. I will tell of the decree of the Lord: He said to me, "You are my Son; today I have begotten you." Begotten not made.

Or Psalm 2,12: Kiss the Son, lest he be angry, and you perish in the way, when his wrath is kindled but a little. Blessed are all those who put their trust in him. The Son has divine power to destroy or bless.

Or Psalm 33,6: By the Word of the Lord were the heavens made; and all the host of them by the breath of his mouth. The Word is creator and has a mouth himself.

Or Psalm 45,6-7: Your throne, O God, endures forever and ever. Your royal scepter is a scepter of equity; you love righteousness and hate wickedness. Therefore God, your God, has anointed you with the oil of gladness beyond your companions. So God himself is King but at the same time anointed by a separate person who nonetheless is God i.e. the Messiah is "God from God."

The same can be concluded from Psalm 110,1 (which Jesus Himself used to silence the Pharisees in Matthew 22,41-46): The Lord said to my Lord, Sit at My right hand, Till I make Your enemies Your footstool.

Proverbs 30,4 mentions God's Son as well: Who has gone up to heaven and come down? Who has gathered up the wind in the hollow of his hands? Who has wrapped up the waters in his cloak? Who has established all the ends of the earth? What is his name, and the name of his son? Tell me if you know!

Wisdom 18,15-16 Your all-powerful Word leaped from heaven, from the royal throne, into the midst of the land that was doomed, a stern warrior carrying the sharp sword of your authentic command, and stood and filled all things with death, and touched heaven while standing on the earth. The Word is a warrior connecting heaven and earth.

Isaiah 7,14 Therefore the Lord himself will give you a sign. Look, the young woman is with child and shall bear a son, and shall name him Immanuel, God with us. The Messiah will be God Himself.

Isaiah 9,6 confirms that the Messiah has divine titles: For to us a child is born, to us a son is given, and the government will be on his shoulders. And he will be called Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace.

The epic of the suffering servant in Isaiah 52,13-53,12 prophecying Jesus' crucifixion, death and final victory: See, my servant shall prosper; he shall be exalted and lifted up, and shall be very high. ... so marred was his appearance, beyond human semblance, and his form beyond that of mortals ... And to whom has the arm of the Lord been revealed? .... a man of suffering and acquainted with infirmity; ...  yet we accounted him stricken, struck down by God, and afflicted. ... upon him was the punishment that made us whole, and by his bruises we are healed. ... like a lamb that is led to the slaughter, ... so he did not open his mouth. ... Who could have imagined his future? ... They made his grave with the wicked and his tomb with the rich, ... he shall see his offspring, and shall prolong his days; through him the will of the Lord shall prosper. ... The righteous one, my servant, shall make many righteous, ... he shall divide the spoil with the strong; because he poured out himself to death. 
cf. Isaiah 2,17 and the Lord alone will be exalted on that day.

& Isaiah 45,25 In the Lord all the offspring of Israel shall triumph and glory

& Isaiah 59,16 He saw that there was no one, and was appalled that there was no one to intervene; so his own arm brought him victory, and his righteousness upheld him. So the Exalted, Redeeming and Triumphing Servant is God Himself.

Jeremiah 23,5-6 foreshadowed the same: The days are coming, declares the Lord, when I will raise up for David a righteous Branch, a King who will reign  wisely  and do what is just and right in the land. In his days Judah will be and Israel will live in safety. This is the name by which he will be called: The Lord Our Righteous Savior.

Jer 31:22 God will make a new thing, a woman shall encompass a man! implicit incarnation...

Daniel 7,9 As I looked, thrones were set in place, and the Ancient of Days took his seat. There's more than 1 throne.

The prophecy of Daniel 7,13-14, the origin of the title "Son of Man" which Jesus claimed for Himself: I saw in the night visions, and behold, with the clouds of heaven there came one like a son of man, and he came to the Ancient of Days and was presented before him. And to him was given dominion and glory and a kingdom, that all peoples, nations, and languages should serve him; his dominion is an everlasting dominion, which shall not pass away, and his kingdom one that shall not be destroyed. So this man has an everlasting dominion just like God Himself (cf. Psalm 145,13).

Zechariah 2,11-12 Shout and be glad, Daughter Zion. For I am coming, and I will live among you, declares the Lord. Many nations will be joined with the Lord in that day and will become my people. I will live among you and you will know that the Lord Almighty has sent me to you. The Messiah is the Lord incarnate sent by the Almighty. 

Zechariah 4,8-10 Then the Word of the Lord came to me: ... Then you will know that the Lord Almighty has sent me to you. The Word speaks and is sent by the Lord.

Zechariah 9,9+14,4 Rejoice greatly, Daughter Zion! Shout, Daughter Jerusalem! See, your king comes to you, righteous and victorious, lowly and riding on a donkey, on a colt, the foal of a donkey + On that day his feet will stand on the Mount of Olives, east of Jerusalem, and the Mount of Olives will be split in two from east to west, forming a great valley, with half of the mountain moving north and half moving south. 
cf. Zechariah 14,9 The Lord will be king over the whole earth. On that day there will be one Lord, and his Name the only name. The Messiah-King is the Lord Himself.

N.B. the title Son of God not necessarily implies being divine. For example Jacob and Solomo in the Old Testament are referred to as Sons of God as well on the basis of their descent from Adam, man created in God's image and likeness (Genesis 1,26). So only when divine properties are associated with the title is it a proper prefiguration of Jesus as "God's only begotten Son" (John 3,16).