Understanding the relationship between the Old Testament and the New Testament is one of the most challenging (but important!) questions a Christian can wrestle with. One of the key texts in thinking about this question is 1 Peter 1:10-12
Concerning this salvation, the prophets who prophesied about the grace that was to be yours searched and inquired carefully, 11 inquiring what person or time the Spirit of Christ in them was indicating when he predicted the sufferings of Christ and the subsequent glories. 12 It was revealed to them that they were serving not themselves but you, in the things that have now been announced to you through those who preached the good news to you by the Holy Spirit sent from heaven, things into which angels long to look.
This passage makes it clear that the message of the of the prophets in the Old Testament is the same as the gospel message proclaimed in the New Testament. In a sermon I preached on this passage, here is an analogy I used to try to illustrate this:
When I was kid there was an old 12” black and white TV that I used to occasionally watch football games on when my dad was watching something else. The reception wasn’t very good and the picture was kind of grainy. But you could still make out the players and follow the game at a basic level. Today in my basement I have a 42” high definition flat screen TV. The difference is stunning—the colors are bright and crisp, you can see so much more of the field, and at times it feels like you are actually there at the game. But if you were to watch the game on the 12” black and white TV while I watched it on the 42” HDTV, we are still watching the same game. We would both be able to tell you who played, what the score was, and what the key plays were. The same is true with respect to the gospel. When we read the Old Testament it is like seeing the gospel in black and white—you can see the basic message, who the key player is and what the key plays are. But when we read the New Testament it is like seeing the gospel in full high definition color—in addition to the basic message, the key player and the key plays you can also see the amazing details that help compose that picture. You can see the fuller scope of the message as well as its background.
No analogy is perfect, but I believe this analogy helps explain how the OT relates to the NT.