John 5:38-39;
38But you do not have His word abiding in you, because whom He sent, Him you do not believe. 39You search the Scriptures, for in them you think you have eternal life; and these are they which testify of Me.
Luke 24:44-45;
44Then He said to them, "These are the words which I spoke to you while I was still with you, that all things must be fulfilled which were written in the Law of Moses and the Prophets and the Psalms concerning Me." 45And He opened their understanding, that they might comprehend the Scriptures.
The New Testament is the fulfillment of the Old. All the Law and the Prophets spoke of and pointed to Christ.
As Paul wrote, by Jesus being crucified, He was "Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross" (Colossians 2:14).
This means that the New Testament fulfilled the Old, just as you say, so I don't think we disagree on this. All I was pointing out by saying you need to read the first half to understand the rest, is that from reading the Old Testament, you gain valuable insight in order to fully understand the New Testament.
"The New Testament is the fulfillment of the Old. All the Law and the Prophets spoke of and pointed to Christ."
But not all the OT has been fulfilled to this day. While it is true that the coming of Christ fulfilled the Mosaic Law and did away with it, there is still plenty of OT prophecies which have not come to pass yet.