Do Jews and Christians both follow the teachings of the Old Testament?
The Old Testament was given solely to the nation of Israel. "And Moses called all Israel, and said to them: "Hear, O Israel, the statutes and judgments which I speak in your hearing today, that you may learn them and be careful to observe them. The LORD our God made a covenant with us in Horeb. The LORD did not make this covenant with our fathers, but with us, those who are here today, all of us who are alive" (Deuteronomy 5:1-3). Moses then reviews the Ten Commandments and the rest of the laws with the people of Israel. Earlier, in building up to this point, Moses said, "And what great nation is there that has such statutes and righteous judgments as are in all this law which I set before you this day?" (Deuteronomy 4:8). The Law contained in the Old Testament was unique to the nation of Israel.
This law ended with the death of Jesus Christ, allowing both Jews and Gentiles to come under one new law. "Therefore remember that you, once Gentiles in the flesh--who are called Uncircumcision by what is called the Circumcision made in the flesh by hands-- that at that time you were without Christ, being aliens from the commonwealth of Israel and strangers from the covenants of promise, having no hope and without God in the world. But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. For He Himself is our peace, who has made both one, and has broken down the middle wall of separation, having abolished in His flesh the enmity, that is, the law of commandments contained in ordinances, so as to create in Himself one new man from the two, thus making peace, and that He might reconcile them both to God in one body through the cross, thereby putting to death the enmity. And He came and preached peace to you who were afar off and to those who were near. For through Him we both have access by one Spirit to the Father. Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone, in whom the whole building, being joined together, grows into a holy temple in the Lord, in whom you also are being built together for a dwelling place of God in the Spirit" (Ephesians 2:11-22). There is no longer a separate law for the Jews and for the Gentiles, but one law for all the nations of the earth. "There is neither Jew nor Greek, there is neither slave nor free, there is neither male nor female; for you are all one in Christ Jesus" (Galatians 3:28). This one law, the gospel, saves all people. "For I am not ashamed of the gospel of Christ, for it is the power of God to salvation for everyone who believes, for the Jew first and also for the Greek" (Romans 1:16). Hence, no one is under the Old Covenant -- neither Jews nor Gentiles. It is solely by Jesus' teachings that we will be judged. "He who rejects Me, and does not receive My words, has that which judges him--the word that I have spoken will judge him in the last day" (John 12:48).
While Christians are no longer under the Old Law, it remains of use to every Christian. "For whatever things were written before were written for our learning, that we through the patience and comfort of the Scriptures might have hope" (Romans 15:4).