There are three things we should understand about the Old Testament and the early Church if we want to read the Scriptures the way the first Christians did. First is that, for the early Church, the Old Testament constituted the Scriptures. The Gospels and Epistles had not yet been written. It was not until about the AD 50s that the epistles began to be written, and the 60s that the Gospel writers began their work. But the Church had the Old Testament, and their faith was that it was all about Jesus, because that is what Jesus had taught his disciples, and they found Jesus and the gospel all throughout. Even when the New Testament writings came along, the Old Testament remained indispensable for the Church in her understanding of Christ.
The second thing to realize is that the Old Testament the early Church used was not the Hebrew version but the Septuagint, also known as the LXX, an ancient translation of the Hebrew text into Greek. This was the common form of the Scriptures used among the scattered Jews because, in their exile, a great many of them did not know Hebrew. It was natural then for the early Church, even though the first Christians were Jewish, to read and study and incorporate this Greek version into their liturgies. And whenever the New Testament writers cited or quoted the Old Testament Scriptures, as they very often did, it was the Greek version that they used. The Septuagint deserves much more respect than many evangelicals have been willing to give it.
The third thing we need to recognize is that, though the early Church found Christ everywhere in the Old Testament, they did not do so by a literal or grammatical method of interpretation but by allegorical or figurative readings of the Scriptures. We can observe this in how the New Testament writers handled the Old Testament. Whenever they cited, quoted or alluded to the Scriptures, the take-away for them was always about Christ and the gospel, even when a literal reading of those Scriptures would have shown no evidence of such — indeed, there are several Old Testament passages which, if taken literally, would be a contradiction of Christ and the gospel. For several centuries, the early Church followed the exegetical pattern of the apostles and New Testament writers. They did not go to the Old Testament Scriptures for history lessons but for spiritual nourishment, the testimony of Christ they contained — and they did not lack for food.