The gospel is the storyline of the bible. It is not something that God decided to try after the Old Testament didn't work
The LORD God said to the serpent, “Because you have done this, cursed are you above all livestock and above all beasts of the field; on your belly you shall go, and dust you shall eat all the days of your life. I will put enmity between you and the woman, and between your offspring and her offspring; he shall bruise your head, and you shall bruise his heel.”(Genesis 3:14-15 ESV)
Immediately after the first sin, God prophesies about the gospel. Talking to the serpent, God speaks about an offspring of a woman who, although he would have his heel bruised, will ultimately crush Satan's head. Jesus was born to a virgin and, though he died in apparent weakness, conquered Satan, sin and death, through his death and resurrection on the cross
“I am the LORD your God, who brought you out of the land of Egypt, out of the house of slavery.(Exodus 20:2 ESV)
As God is about to give the 10 commandments to Israel, he reminds them that he has brought them out of slavery, he has rescued them. The law was not a way of earning God’s favour so that he would bring them out of slavery, He had already done that. The law was a response to God’s grace, how they should live in the light of God’s rescue. In one of the most well known lists of “rules to follow” of all time, we see that it is God’s grace, not our obedience, which is the important factor.
But he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed.(Isaiah 53:5 ESV)
Isaiah 53 talks about a suffering servant who will come and be our substitute. The transgression and iniquity are ours but, because Jesus was pierced and crushed in our place, we are healed, we are at peace with God.
The next day [John] saw Jesus coming toward him, and said, "Behold, the Lamb of God, who takes away the sin of the world!(John 1:29 ESV)
Then I looked, and I heard around the throne and the living creatures and the elders the voice of many angels, numbering myriads of myriads and thousands of thousands, saying with a loud voice, "Worthy is the Lamb who was slain, to receive power and wealth and wisdom and might and honor and glory and blessing!"(Revelation 5:11-12)
All through the Old Testament, lambs are used as sacrifices to remove the people’s sin. This was an imperfect system that had to repeated over and over again. This system was perfected and completed by Jesus, the Lamb of God who takes away the sin of the world. This is the culmination of God’s work and the angels never get tired of it! We will join with them in the new creation, singing about the lamb who was slain for the rest of eternity!