There are two reasons why it can be beneficial to read between the Old Testament and the New Testament rather than attempting to read straight through the Bible or to read only the New Testament.
First, when trying to read straight through the Bible, readers often bog down when they come to the parts that may not hold their attention, such as the long genealogies found in the first 9 chapters of the Old Testament book of Numbers. Moving between the Old and New Testaments each week helps to keep us reading through the entire Bible when we hit difficult-to-read spots.
Secondly, some Christians today erroneously believe that the Old Testament is not relevant to believers. By moving between the Old and New Testaments, the reader is able to see how closely these sections of the Bible are tied together. There is much to learn about God and ourselves in the Old Testament.