Elohim is a Hebrew word used often in the Pentateuch (Genesis through Deuteronomy) to refer to God. It is one of the Hebrew ways of saying the word "God". Eloah is the singular form for "God". Elohim is the plural form. Yet strangely, Elohim is the form that is used throughout the Pentateuch to refer to the one true "God". The verbs and articles used with Elohim show that it is to be taken as singular, meaning "God" and not "gods", but it is indeed strange that the plural form is used.
Some scholars call this a plural of majesty. The idea is that God is sooooooo big, glorious, and amazing that the singular word "me" is not good enough for Him. Instead, "we" is needed to express His bigness. In that culture, it was not uncommon for kings or people writing about kings to refer to a king in the plural as "we" or "us". Have you heard of "The Royal We"? Its sort of like that. Many scholars believe that this use of a plural "Elohim" does not support the Trinity but instead is a plural of majesty intended to show God's greatness. The problem with this theory is that the Old Testament never uses the plural of majesty to describe any king or other leader either in narration or in speech. If the Bible used this sort of writing technique which was common when speaking of kings, then why is it never used anywhere to speak of kings?
Instead, I believe that this plural use of Elohim is our first glimpse at the Trinity as God reveals Himself to us in His Word. From the very beginning, God used a name for Himself that showed us He is One and Three.
We, Us, Our
The Old Testament also shows the idea of the Trinity in the ways that God speaks to Himself. For example...
Then God said, "Let us make man in our image, in our likeness, and let them rule over the fish of the sea and the birds of the air, over the livestock, over all the earth, and over all the creatures that move along the ground." -- Genesis 1:26
God decides that He is going to make man in His image. Instead of saying "I will make them in myimage," He says "Let us make them in our image." This sort of thing happens again in Genesis 3:22, Genesis 11:7, and Isaiah 6:8. Some scholars would suggest that God is talking to the angels on these occasions. However, the angels are not even mentioned anywhere in the context of any of these passages. In this specific example, God is creating man. We know that the angels were not involved in the act of creation, God alone created the entire universe. It seems that again we have God revealing His Triune nature to us as we see the Trinity communicating with one another.
God with God
Your throne, O God, will last for ever and ever; a scepter of justice will be the scepter of your kingdom. You love righteousness and hate wickedness; therefore God, your God, has set you above your companions by anointing you with the oil of joy. -- Psalm 45:6-7
The Old Testament also has the occasional scene like the one above. In these scenes, God plays more than one character at the same time. In the passage above, we have the author of the Psalm speaking to God and saying good things about Him. Then the Psalmist tells God that the God of God is going to set God above all others. Confusing? Take a look at Psalm 110:1, Malachi 3:1-2, and Hosea 1:7. You will see the same sort of thing going on. The only way that these passages make any sense, is if God is one Being who exists in more than one Person.
References to the Holy Spirit as distinct
The Old Testament also gives us evidence that the Holy Spirit is God but is also somehow distinct from God.
In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. And God said, "Let there be light," and there was light. Genesis 1:1-3
Here it looks like the Holy Spirit is distinct from God. God created the heavens and the earth. The Holy Spirit hovers over the waters. Then we are back to God speaking light into existence. In Isaiah 48:16 and 61:1, the Spirit of the Lord and the Lord seem to be distinct as well. Again, the Old Testament is teaching a plurality in the Godhead (Godhead is a fancy theology word to refer to the Trinity)
The angel of the Lord
Finally, the Old Testament makes many references to a character called "THE angel of the Lord". I put the word "THE" in caps because it isn't just "an" angel, its "THE" angel. If I were to mention "a" woman in my life, you might think I was talking about my mom, my daughter, my wife, my grandmother or any other woman in my life. But if I mentioned "THE" woman in my life, you would know I meant Christy (my wife). When you hear the word "angel," you probably get an image of a glowing winged human with white robes. But the word "angel" actually only means "messenger". The Old Testament makes many mentions of various angels or messengers of the Lord. However, there seems to be one such messenger that is special. This is "THE angel of the Lord".
When Hagar runs from Sarah in Genesis 16, it is "THE angel of the Lord" that shows up to speak with her. Look at how Hagar talks to this special angel/messenger.
She gave this name to the LORD who spoke to her: "You are the God who sees me," for she said, "I have now seen the One who sees me." -- Genesis 16:13
She calls this angel "LORD" (literally Yahweh) and then "God"! That's not just any old angel! You see this kind of thing as well in Exodus 3:2-6, Exodus 23:20-22, Numbers 22:35, Judges 2:1-2, and Judges 6:11, 14. The angel of the Lord is given worship and recognized as God. Some scholars believe that this is the second member of the Trinity, the Son of God, before He took on flesh and was known to us as the God-Man Jesus. Those who believe this (like me) call these Old Testament appearances of Jesus "theophanies".
Either way, we see that there is a character in the Old Testament who is called THE Messenger from God, but this Messenger also is God! God sends Himself sometimes to get His messages out. Interesting.
Does the Old Testament come right out and say that God is one Being who eternally existed as three Persons? Nope. However, the Old Testament is filled with numerous strange passages that only make sense if God is indeed Triune. The Trinity is indeed in the Old Testament!