Hi there, yes I can see you are really asking the question, and there are a lot of people who ask but don't really want to know. So I appreciate that you are being tactful and polite.
It's not a matter of throwing out the old testament. It's necessary and indispensible.
What you are asking is a pretty complicated question that could be answered in many different ways, but I will try to give you are short and simple version for convenience here on Yahoo.
The new testament does not cancel out the old testament. It fulfills it. In the beginning, Genesis 4:27 says "At that time men began to call on the name of the LORD." Later on, Abraham was friends with God, and had faith in Him, which pleased God and so He made a covenant with Abraham about his descendants. For this, Abraham is called the forefather of faith. Later still, God rescued Abrahams descendants, the Israelites, from slavery in Egypt, and used Moses to reveal His laws to them, as the 10 commandments, and led them to the promised land, fulfilling His covenant with Abraham.
The Israelites were given the law, but they discovered that the law did not set them free, rather it convicted them of guilt. They broke the commandments over and over again, and were punished for their wickedness. They came to realize that they were slaves to sin, as God wished to show them, and used them to make an example for us. So they asked for a messiah to save them and release them from the trap of sin and the death sentence they recieved simply for being born on this earth.
So God sent Jesus and made a new covenant with all people. Jesus fulfilled all the laws perfectly, and made atonement once and for all for all the sins of the people everywhere, so long as they acknowledge that his sacrifice releases them from the burden of sin. He showed that although we are prisoners of sin here in this life, there is a resurrection and there will be a new creation where we will be free from death, tears, and sin. Jesus came once to atone for sin, and promised that he would come again to lead the way to salvation that would release us from sin and allow us to enter the kingdom of Heaven. This has been Gods plan all along, but the creation work proceeds in stages, so that we can be bonded with God through blood (bread and wine) and suffering, and make our love for Him pure, so that we can be adopted as God's children and become His inheritors.
So, to reiterate quickly:
Adam and Eve sinned, and were punished for it, cast out into the world full of suffering and death
Sometime later people began to call on God for help.
Much later Abraham put his faith in God, and was rewarded for it.
Later still, God used Moses to free the Israelites from slavery, reveal his laws to the people, and lead them to the homeland He promised them
But they were still not free from suffering and evil. They broke all of Gods laws and suffered the consequences.
Jesus came and showed the way to fulfill the law instead of being convicted by it, to live by love, mercy, repentance, humility, and faith, and thereby allow the gates of Heaven to be opened for all people.
Jesus made atonement for sin, showed us the resurrection, and promised to return to release us from the power of death and lead us to salvation.
I hope that helps you understand a little better. It's not that the old covenant doesn't matter anymore, it's more like we have a better deal available to us. The old covenant led to death and sin, the new covenant sets us free. We don't throw out parts of the Bible, we use them as God wants us to; as examples that show us where we should put our hearts, out tongues, and our intentions.