Jesus died for our sins “according to the scriptures.” What scriptures in particular?

Paul states in 1 Corinthians 15:1-4 that Jesus died for our sins according to the scriptures. I see that in Paul’s epistles but where is that anywhere else in the Bible or is that part of the revelations that Paul received. I do understand that the age of grace is a mystery but I don’t understand why so many churches think the church started on Pentecost or in Matthew when Jesus started picking disciples?  

I think I understand your question.  Consider the following:

Even the events of death, burial and resurrection, were historically predicted in the Old Testament as I Corinthians 15:3,4 say: 

“For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures;  And that he was buried, and that he rose again the third day according to the scriptures.”

The elements of the gospel Paul preached was not a mystery.

The death of Christ was according to the scriptures in Isaiah 53:5,6:

“But he was wounded for our transgressions, he was bruised for our iniquities: the chastisement of our peace was upon him; and with his stripes we are healed. All we like sheep have gone astray; we have turned every one to his own way; and the LORD hath laid on him the iniquity of us all.”

The burial of Jesus Christ was predicted by he himself in Matthew 12:39,40

“But he answered and said unto them, An evil and adulterous generation seeketh after a sign; and there shall no sign be given to it, but the sign of the prophet Jonas: For as Jonas was three days and three nights in the whale’s belly; so shall the Son of man be three days and three nights in the heart of the earth.”

And the Resurrection of Christ was written in Psalms 16:9,10

“Therefore my heart is glad, and my glory rejoiceth: my flesh also shall rest in hope. For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption.”

However, even though the elements of the gospel was not a mystery, what was a mystery is the details of how the cross saved sinners, that God can now justify sinners by faith.  That God would dispense his grace in this age to Gentiles apart from Israel and the fulfillment of their promises and through the fall of Israel, form out of all nations the Body of Christ. This is what Paul calls “the mystery of the gospel” in Ephesians 6:19.

So Paul used the Old Testament scriptures to teach that Jesus is the Christ and that he died, was buried and rose again as the scriptures said.  Then he could teach his revelation concerning all that the cross accomplished and the grace of God to us Gentiles. Hope this helps.