There are people who actually ignore the Old Testament part of the Bible and read only the New Testament. Without the OT we can't properly understand the NT! The NT is built on the foundation of the OT. But at the same time we must know that the concept of the old and new parts of the Bible is different from the old and new covenants which God made. The old covenant made through Moses is now obsolete (He.8:13)! What is now operating is the new covenant which God made through the blood of Jesus on the cross. The old covenant existed even when Jesus lived on the earth and the new came only when He died.
The heart of the old covenant was the Law, with the condition that God would bless those who kept the Law and curse the disobedient (Deut.28). Obviously, no one except Jesus was able to keep the Law fully (Ro.3:20,23). One of the reasons why God gave the Law was that everyone would get to see and admit that they had sinned (v.19). Once a person comes to this understanding, he will cry out to God for mercy, and that is when the new covenant will become attractive to him (He.8:12). Under this covenant, God will give such a man a new heart, write His laws on his heart and help him to walk in His ways (Ez.36:26,27). This man now wants to stop sinning and live in obedience to the Lord.
The old covenant was a major part of God's overall plan for man's salvation. Without it, God could not go on to the new. The old was like a guardian looking after a child till he came of age, able to take up his responsibilities (Ga.3:24,25). Once he acquires this sense of responsibility he doesn't need a law any more telling him to avoid sin under threat of curses! He himself wants to be pleasing to the Lord.
Before we can come to the place in our practical life where we are 'free from the Law' (Ro.7:6), we should know what it is to find ourselves helpless under the Law. That is how we can know what unmerited favour (grace) really means for God and us. This is the antidote for the error that exaggerates 'grace' without realising the serious problem of sin. And it is the Old Testament that helps us to learn this, by looking at what God asked Israel to do and how they failed, and seeing how we ourselves fail when we try to obey God in everything. Then we learn to go to God for both aspects of grace – mercy and help (He.4:16). We also see many sides of God which are not obvious from the New Testament.
We certainly need the OT. But we need to remember that we are not under the old covenant but the new. We are free from the Law, and we mustn't be looking now for its blessings and curses! We are under the law of the Spirit of Christ (Ro.8:2) which transforms us inwardly. The greatest blessing now is that we can be victorious over temptation and become more like Jesus. We shouldn't miss this, by getting taken up with the earth-based blessings of the old covenant. Seeking godliness has to be first, and then God will give other things too.