According to Matthew 5:17, Jesus says that He came not to get rid of the Law or the Prophets (i.e. the Old Testament) but to fulfill them. The other writers of the New Testament agree with this.
For example, in Romans 3:25-26, Paul writes "God presented [Jesus] Christ as a sacrifice of atonement, through the shedding of His blood - to be received by faith. He did this to demonstrate His righteousness, because in His forbearance He had left the sins committed beforehand unpunished - He did it to demonstrate His righteousness at the present time, so as to be just and the One who justifies those who have faith in Jesus." Hebrews 7:27 says "Unlike the other high priests, [Jesus] does not need to sacrifice day after day, first for His own sins, and then for the sins of the people. He sacrificed for their sins once for all when He offered Himself." From these verses (and similar passages) we see that Jesus fulfilled the Law of atonement of sin; He did not get rid of it but fulfilled it by being the ultimate sacrifice for us. Ephesians 5:2 shows that Jesus' sacrifice was acceptable to God, thus being able to atone for our sin. (In case you want to know why Jesus needed to be sacrificed only once, read Hebrews 10:1-18)
Throughout the Gospels, we see the authors pointing out how Jesus fulfilled the prophecies of the Prophets. The author of Matthew points it out more than the other 3. We also see the authors of the rest of the New Testament pointing it out also all throughout their writings. Here is a table filled with prophecies from the Old Testmant that Jesus fulfilled. Feel free to examine it, but for now I will mention one as an example.
In the Old Testament, the prophet Micah said that out of Bethlehem will come for God one who will be ruler over Israel, whose origins are from of old, from ancient times. (Micah 5:2) As seen in Matthew 2:1-6 and Luke 2:4-7, Jesus was born in Bethlehem. According to John 1:1-5 and 1:14, Jesus existed in the beginning and became a human, dwelling with us. Even Jesus Himself claims He existed before even Abraham in John 8:58. From the genealogies of Jesus, though they may be confusing, they both agree that Jesus came from the line of King David, thus having royal blood. While being questioned by Pilate (John 18:33-38), Jesus was asked if He was a king, with Jesus saying that His kingdom was not of this world. And in the future, as recorded in Revelation, Jesus will come again set up His kingdom on earth, ruling from Jerusalem. So we see that Jesus fulfilled this prophecy from the Prophet Micah. This is just one of over 300 prophecies that Jesus fulfilled.
Christians in a way still obey the Old Testament, but they do so through Jesus. Because Jesus fulfilled them, we are living in obedience to them by living for Jesus and having our faith in Him.