These are just some OT verses and passages I've found that interestingly hint at what comes in the NT (besides the prophecies of course). Hopefully I can add more over time.
If you want a free online Bible to follow along with, BibleHub is a good one with many versions - my fav is BSB.
A Plan to Save
II Samuel 14:14 - this is an amazing passage in that while recognizing the inevitabilty of death, says God comes up with a plan to overcome that!
A favorite of mine is Ecclesiastes 3:11 that indicates eternity is the plan but we may not know yet how God carries it out.
Jeremiah 29:11 - probably most regular Bible readers already know of this verse, but again emphasizes how God is planning for His people to have a future and a hope!
Genesis 22 - the story of Abraham willing to sacrifice his only son. This was the son he had waited on for a long time. Yet he obeyed God and had faith that God would provide the sacrifice. Abraham trusted in God's plan even though it seemed like it might mean losing everything he had waited for.
Doing right greater than legal requirements
Micah 6:8 - all that's required is to do justly, love mercy, and walk humbly
Habakkuk 2:4 - the righteous will live by faith, an early hint of faith over legal system
Psalm 51:10-13 - another passage probably well known to regular Bible readers, it emphasizes NT concepts like clean heart and right spirit (inside), the joy of His salvation, and bringing sinners back to God!
Hosea 6:6 - another verse promoting mercy over a sacrificial system.
Wise hints at the NT
Proverbs 22:19 - wise to keep one's trust in the LORD!
Proverbs 11:30 - following the Great Commission is wise before there is a great commission!
Ecclesiastes 12:11 - another amazing wisdom tie-in as this one associates The Shepherd with carpentry!
Proverbs 3:5,6 - another well known couple of verses emphasizing trusting God and doing so with all your heart.
Proverbs 28:13 - an early look at confessing with repentance to find mercy.