Why should a Catholic (or any Christian) know the Old Testament? I'll give you four reasons. Four Scriptural reasons. Just use this simple mnemonic: Matthew, Mark, Luke, and John.
Ok, it's not actually a mnemonic. But in those four names (which you know so well) are the four reasons. What do I mean? Well, you needn't go further than five verses into any of the Gospels before you come across a reference to the Old Testament!
Who are these people? (Matthew 1:1-16)
 The book of the genealogy of Jesus Christ, the son of David, the son of Abraham.The first line of St. Matthew's Gospel presents us with three questions: who is David, who is Abraham, and why does it matter? The next fifteen verses answer these questions, but only if you are familiar with the Old Testament. The genealogy provided by the author of the Gospel is saturated with the Old Testament.
Abraham was the father of Isaac, and Isaac the father of Jacob, and Jacob the father of Judah and his brothers,  and Judah the father of Perez and Zerah by Tamar, and Perez the father of Hezron, and Hezron the father of Ram,  and Ram the father of Amminadab, and Amminadab the father of Nahshon, and Nahshon the father of Salmon,  and Salmon the father of Boaz by Rahab, and Boaz the father of Obed by Ruth, and Obed the father of Jesse,  and Jesse the father of David the king.
And David was the father of Solomon by the wife of Uriah,  and Solomon the father of Rehoboam, and Rehoboam the father of Abijah, and Abijah the father of Asa,  and Asa the father of Jehoshaphat, and Jehoshaphat the father of Joram, and Joram the father of Uzziah,  and Uzziah the father of Jotham, and Jotham the father of Ahaz, and Ahaz the father of Hezekiah,  and Hezekiah the father of Manasseh, and Manasseh the father of Amos, and Amos the father of Josiah,  and Josiah the father of Jechoniah and his brothers, at the time of the deportation to Babylon.
 And after the deportation to Babylon: Jechoniah was the father of She-alti-el, and She-alti-el the father of Zerubbabel,  and Zerubbabel the father of Abiud, and Abiud the father of Eliakim, and Eliakim the father of Azor,  and Azor the father of Zadok, and Zadok the father of Achim, and Achim the father of Eliud,  and Eliud the father of Eleazar, and Eleazar the father of Matthan, and Matthan the father of Jacob,  and Jacob the father of Joseph the husband of Mary, of whom Jesus was born, who is called Christ.
Matthew tells us of the patriarchs: Abraham, Isaac, and Jacob. Matthew mentions four women who were ancestors of the Christ: Tamar, Rahab, Ruth, and Bathsheba (the wife of Uriah). Why does Matthew mention them? Because of the peculiarity of their circumstances:
Tamar was married to Judah's son Er (who died), so Judah gave her another of his sons, Onan, as a husband (who also died). Judah promised her another of his sons as husband, Shelah, but Judah sent her away and never fulfilled his promise (for fear that Shelah would also die). So Tamar disguised herself as a harlot, and Judah went into her and she conceived twins (Perez and Zerah). When this was all revealed to Judah, he was greatly ashamed. Yet it was through this pseudo-harlotry that Perez, a forefather of Jesus Christ, was born. (cf. Genesis 38)These women, these unions, should be an embarrassing blot on a family tree! What king would be open about being the son of numerous harlots and pagan foreigners? What king would gladly claim the throne of his adulterous murderous ancestor? This (along with the crucifixion) is a stumbling block to Jews, and folly to the Gentiles. Yet Matthew records the power of God to take the lowly and the sinful and exalt them, glorifying Himself in the process.
Rahab was a pagan harlot from Jericho who aided the Israelites spies sent into the city by Joshua. Because of her fidelity to Israel, she and her family were permitted to live when the Israelites captured the city. For her faithfulness, she is mentioned in the "Hall of Fame" in Hebrews 11. (cf. Joshua 2, 6)
Ruth was a Moabite, the daughter-in-law of Naomi (wife of Elimelech, of Bethlehem in Judah). When Ruth's husband died, Ruth stayed with Naomi, telling her "where you go I will go, and where you lodge I will lodge; your people shall be my people, and your God my God." This was a powerful statement in a time when Israel continually rejected the Lord as God. (cf. Judges) Ruth eventually married Boaz, a kinsman of her father-in-law. These had a son, Obed; Obed was the father of Jesse; Jesse was the father of King David. Thus another pagan foreigner became an ancestor of Jesus Christ, in whom the dividing wall of enmity between Jew and Gentile was torn down. (cf. Ruth 1-4)
Finally, Bathsheba was the wife of Uriah, a man of King David's army. David was enamored with Bathsheba's beauty and she conceived a son by him. To hide this sin, David tried to get Uriah to sleep with Bathsheba. When this plan failed, David had Uriah slain on the battlefield. Because of these two grievous sins — adultery leading to murder — the Lord took the child to Himself. David comforted Bathsheba afterwards, and she born another son, Solomon the wise, who would be an ancestor of Jesus. (cf. 1 Samuel 11-12)
Matthew makes it clear that Jesus is royalty: he is the descendant of David, King of Israel! (The significance of this is made clear especially in Luke 1:32-33.) Matthew brings in the sordid history of Israel: her deportation and exile. Finally, Matthew draws attention to Joseph and Mary: he does not say "Joseph the father of Jesus," but rather "Joseph the husband of Mary, of whom Jesus was born."
It is clear from the beginning of Matthew's Gospel that he wrote with the expectation that his readers would know who these figures in Israel's history were and why they were significant to be ancestors of the Messiah. That is the first reason to know the Old Testament: to know who these people are, so that we know where Jesus came from.
Isaiah the what? (Mark 1:1-4)
 The beginning of the gospel of Jesus Christ, the Son of God.  As it is written in Isaiah the prophet, "Behold, I send my messenger before thy face, who shall prepare thy way;  the voice of one crying in the wilderness: Prepare the way of the Lord, make his paths straight—"St. Mark gets right to the point. Jesus is the Messiah and the Son of God. But Mark does not begin by recording Jesus' origin; rather, he tells us of the man who came before him, John the Baptist. But Mark sets the stage for John the Baptist by invoking Isaiah the prophet. If we don't know the Old Testament, we won't know 1) what a prophet is, 2) who Isaiah was and when he lived, and 3) what he prophesied.
 John the baptizer appeared in the wilderness, preaching a baptism of repentance for the forgiveness of sins.
So much of the Gospels — essentially all of Christ's life, ministry, Passion, death, and Resurrection — is the fulfillment of Scripture. No less than a dozen times does Matthew inform his reader that some action of Christ was done "to fulfill what was spoken" by some prophet. Luke records that, on the road to Emmaus, Christ interpreted "all the scriptures the things concerning himself," and afterwards told the Apostles that "everything written about me in the law of Moses and the prophets and the psalms must be fulfilled." That is the second reason to know the Old Testament: to realize that Jesus Christ is the fulfillment of numerous prophecies.
Herod, Abijah, and Aaron, oh my! (Luke 1:5)
 In the days of Herod, king of Judea, there was a priest named Zechariah, of the division of Abijah; and he had a wife of the daughters of Aaron, and her name was Elizabeth.What is Judea? Who is Herod? What's a priest? Who was Abijah? Who was Aaron? This is more than simply knowing who these people are, this is knowing the cultural setting into which Jesus was born.
Israel was a kingdom... but Herod was not a true Davidic King. Israel had been split in two during the time of Solomon's son Rehoboam, and this wound was never fully healed. Judea (i.e. Judah) was the southern Kingdom, home to Jerusalem, where David and Solomon had reigned over the whole Kingdom of Israel for a time.
Zechariah (the father of John the Baptist) was a priest. The priestly men of Israel were divided into three groups: the high priest (a descendant of Aaron), the priests (other descendants of Aaron), and their ministers (other descendants of Levi, the tribe to which Aaron belonged). These Aaronic priests were assigned by King David according to twenty-four divisions, as recorded in 1 Chronicles 24:1-18. Here we find that Abijah's division was the eighth. Both Zechariah and Elizabeth are descendants of Aaron. As for what a priest does, the Old Testament book of Leviticus ("pertaining to the Levites") describes that in detail.
This single verse gives us a historical setting: Herod was king of Judea in a certain period of time, and Zechariah was serving in the Temple during two specific weeks of the year. This verse also gives us a cultural setting: a nation, a kingdom, with a priesthood. This is the third reason to know the Old Testament: to understand the cultural and historical circumstances into which Jesus was born.
Another "beginning"? (John 1:1)
 In the beginning was the Word, and the Word was with God, and the Word was God.Do these words sound a little familiar? Wasn't there some other "beginning" mentioned somewhere in Scripture? Yes.. yes... back in Genesis. The first verse of the Old Testament is recalled by the first verse of the Gospel of John.
John's Gospel is full of spiritual parallels of the Old Testament. In John 4, when Jesus is speaking with the Samaritan woman at the well, he records God's unfolding of a historical event (the dilution of northern Israel — Samaria — by five pagan nations by King Sargon, described in 2 Kings 17:24) in the personal encounter between two people. Jesus asks the woman to bring her husband, and the woman says she has no husband. Jesus replies that she has spoken truthfully, because she (representing Samaria) has had five husbands (representing Babylon, Cuthah, Avva, Hamath, and Sephar-vaim) and that the one she is with now (Jesus, the Lord) is not her husband. This hearkens back to the book of the Prophet Hosea as well.
This does not mean that the Old Testament event (the population of Samaria by pagan nations) nor the New Testament event (the encounter between Jesus and the Samaritan woman) were not actual historical events; but it shows that God planned the historical event of the Old Testament to be a sign pointing to a deeper spiritual reality to be fulfilled and manifested in the New Testament. St. Paul calls the story of Sarah and Hagar an allegory. By this, he does not mean the story of them in Genesis is made up, but that by God's design, it points to something deeper, something which is only made manifest in the light of Christ.
When the authors of the New Testament books wrote in ways that evoke the Old Testament, they did so because God directed them! Without knowing the history of Israel, all the way back to Abraham, Noah, and even Adam, the New Testament lacks a critical dimension. As St. Augustine said, the New Testament is concealed in the Old, and Old Testament is revealed in the New. This is the fourth reason to know the Old Testament: to see it revealed in greater spiritual depth in the New Testament, through Jesus Christ.
I hope these four reasons inspire you to reconsider just how important the Old Testament scriptures are to you and to your faith. If you are interested in learning more about the Old Testament, I strongly urge you to find a Great Adventure Bible Timeline study in your area.