Should Christians be studying the Old Testament since the New Testament is here?
In the Beginning
You cannot get to the gospel until you get to the reason for the gospel, and that is the fall of mankind in the Garden of Eden. Without knowing the reason for our sin nature, we’d never admit we’re sinners, and we’d never come to God for His mercy and grace. Adam and Eve sinned and were banned from the Garden, and then the earth was cursed. Nothing would ever come easily again, at least not this side of the kingdom. Mankind has decided for himself to choose his own way and has reaped the consequences of those decisions for thousands of years, so if we know what happened in the beginning, we’ll better understand why Christ had to come. Genesis is a good place to begin when reading and studying the Bible, not just the Old Testament. It gives us a solid foundation upon which to build our understanding of redemption. 
For Our Sake
If you read the Old Testament, you’ll find out that God tells it like it is. He shows the heroes and heroines of the faith, warts and all, but the Apostle Paul who knew the Scriptures well, understood that “these things happened to them as an example, but they were written down for our instruction, on whom the end of the ages has come” (1 Cor 10:11). God was thinking ahead for our sake as He says these acts were to “be recorded for a generation to come, so that a people yet to be created may praise the LORD” (Psalm 102:18). We are part of that generation “on whom the end of the ages has come,” so “whatever was written in former days was written for our instruction, that through endurance and through the encouragement of the Scriptures we might have hope” (Rom 15:4), however, we shouldn’t really look at the bad examples and say, “How can they be that way,” but rather ask ourselves, “How am I like that?”
Knowing the History
It’s also good to read the Old Testament to see just how the Jewish customs and traditions have been carried over into the New Testament. In the feast days of the Old Testament, you can see the foreshadowing of Jesus’ coming work at Calvary because they all point to Jesus in significance, and they find their fulfillment in Him. You read your Old Testament and see why the Samaritans are so despised by the Jews. Read where the nation of Israel split into two separate nations, where one became the Northern Kingdom (called Israel) who settled in Samaria, and then the Southern Kingdom with the tribes of Judah, Benjamin, but included the Levites. The Southern Kingdom was called Judah, from which we get the word “Jews.” It’s also interesting to note that on a couple of occasions, Israel and Judah were at war with one another. Reading about Israel’s historical background gives us a much better understanding of our reading the New Testament.
By reading and studying the Old Testament, you can discover the dozens of prophecies that Jesus fulfilled in His coming, His sinless life, His death, and His resurrection. Isaiah 53, Psalm 24, and other Messianic Psalms point directly to the Lamb of God, Jesus Christ. There is little disagreement that the Bible is a “H.I.M.” book. It is all about Him. Even in the beginning, we read about the coming Redeemer Who will come to crush the enemies head, even though His own heel will be bruised (Gen 3:15).
Experience is the best teacher, and we have the advantage of reading many of the saint’s experiences that they had to learn the hard way. They learned the hard way so we wouldn’t have too. I remember seeing friends and family make some very big mistakes (just as I have), but they learned some very valuable lessons (as I have). The Old Testament is full of such lessons. All we have to do is pick it up and read it. Read about Samson’s fall from grace due to his continual pursuit of pleasure. Read about Moses’ bad beginning when he had to escape from Pharaoh’s hand after murdering an Egyptian. The Old Testament has stories of new beginnings, and second and third chances. And you’ll read about God’s unmerited favor and grace toward Lot and his family thanks to Abraham’s intervention. These historical accounts contain some of the richest histories of the patriarchs and saints in the Old Testament. The author of Hebrews saw their value (Heb 11), and I think we should too. The Old Testament is just as inspired as the New Testament is.
One disbeliever told me something that confused me. He said that there is no absolute truth…so how was I supposed to believe what he said? Could he be absolutely sure that there is no absolute truth? There is truth. Jesus Himself is the way, the life, and the truth (John 14:6), and the Old Testament is full of truths. For example, there is so much godly wisdom in the Old Testament that it’s hard to know where to begin. Naturally, “The fear of the LORD is the beginning of wisdom, and the knowledge of the Holy One is insight” (Prov 9:10). Job understood that “the fear of the Lord, that is wisdom, and to turn away from evil is understanding” (Job 28:28). To find that wisdom, find your place in the Old Testament.
A Bible dictionary will be indispensable in your studies of the Old Testament. A good Bible atlas will add a lot to what you’re reading because it shows what the land was like, what the weather and conditions were like, and some even have maps that show the many different nations that surrounded Israel and through the different time periods. When I read the New Testament, I’m surprised at the number of scriptural notations from the Old Testament. Many times the New Testament authors quote Old Testament verses, and frequently they refer to the patriarchs like Abraham (Rom 4; Heb 11). In fact, many Old Testament Scriptures are used to explain the gospel of Jesus Christ. What I do is read some out of the Old Testament and some out of the New Testament every day, and I find that I get more out of the New Testament by reading daily out of the Old Testament. Jesus preached out of the Old Testament. The Apostle Paul referred to “the Scriptures.” The Apostle Peter quoted the Old Testament. That’s because those things were written for us. It’d be a shame not to read them.
Here is some related reading for you: Is the Old Testament Law Still Important Today? 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.