Most Christians appear to have a limited understanding of or appreciation for the Old Testament. The New Testament makes sense to them, but the Old Testament is a mystery, with its diverse kinds of literature and a seeming lack of connection with the New Testament. Most Bible reading plans don't help this very much. Either they read straight through the Bible, in which the New Testament simply follows the Old, or they have some OT and some NT every day, but with no connection drawn between them. Perhaps one way of rectifying this situation is to use the New Testament as something of a search engine for the Old. This has the dual advantage of connecting the testaments and of clarifying the significance of those connections.
If we start with Matthew, the first verse gives us a number of Old Testament connections. "The book of the genealogy of Jesus Christ, the son of David, the son of Abraham". There are three connections right away. "The book of the genealogy" connects us to Genesis 5, which begins "This is the book of the generations of Adam." That genealogy takes us from Adam to Noah. The subsequent genealogy in Genesis 11 takes us from Noah to Abraham. So read Matthew 1, Genesis 5, and Genesis 11. "The son of Abraham" connects us to the story of Abraham in Genesis 12-25. This may seem long for some readers, so read Genesis 12, 15, 17, and 21-22, which will give the reader the substance of Abraham's story and clarify the importance of Abraham for the story of Jesus Christ. "The son of David" takes us to the story of David. The Old Testament gives 61 chapters to the story of David (from 1 Samuel 16-1 Kings 2). That doesn't include the 73 Psalms of David in the Book of Psalms. So read 2 Samuel 1-8, which gives us the best part of the story of David. That includes the covenant that God made with David which issues in the promise of the everlasting Davidic king.
The genealogy of Jesus from Abraham to Jesus also links to a number of other OT passages. First, we have the mention of Judah and Tamar, which takes the reader to Genesis 38. The mention of Rahab (Matthew 1:5) takes us to Joshua 2, with the beginning of the conquest of Canaan. Verse 5 also mentions Ruth, which takes us to the Book of Ruth. The mention of "the wife of Uriah" takes us to 2 Samuel 11-12 which also tells of the birth of Solomon, thus continuing an explanation of the working out of God's promise to David.
The mention of Josiah (verse 11) takes us to 2 Kings 22-25, which tells of the glorious reign of Josiah and the sad demise of the kingdom of Judah. The reference to Zerubbabel in verses 12-13 takes us to the books of Ezra and Nehemiah. The story is summarized in Ezra 1-5.
Matthew 1 ends with the birth of Jesus and the reference to the Immanuel prophecy from Isaiah 7. To get the context for that pronouncement, the reader can read Isaiah 6-12.
From the first chapter of Matthew, the reader has been introduced to many of the major themes and persons of the Old Testament. It has also drawn the reader to more than thirty chapters of the Old Testament, and has given the reader some sense of the importance of those passages in the unfolding story of God's redemption.
A person who has a Bible with cross references can thus easily make connections with OT passages by noting the passages that the NT quotes. When making those connections, it is important to get something of the context for the OT citation. Continuing our use of Matthew, chapter 2 will take the reader to Micah 5 (read chapters 4-6); Hosea 11:1 (read chapters 10-11); Jeremiah 31 (read chapters 30-33). The final reference in Matthew 2 "he would be called a Nazarene" is something of a puzzle, but a reading of 2 Kings 15 and Isaiah 8-9 might go a long way to clearing up the puzzle.
By following the directions given by the New Testament, the reader can, over time, develop a solid understanding of the Old Testament, and the way it relates to the New.