Start in the Old Testament, because that is where the promise starts. Back then the Spirit of God came on particular people for particular tasks — kings, prophets, judges, craftsmen. But notice something: the Spirit came, and the Spirit went. He was not a permanent resident.
That is why, after Saul's disobedience, "the Spirit of the Lord departed from Saul" (1 Sam. 16:14). And it is why David, after his own great failure, prayed a prayer that would make no sense today: "Do not cast me from your presence or take your Holy Spirit from me" (Ps. 51:11). David could lose the Spirit. He knew it.
But woven through the prophets is the promise of a different day — a day when God would pour His Spirit out on everyone and put Him inside people to stay:
"I will pour out my Spirit on all people. Your sons and daughters will prophesy..." — Joel 2:28-29 (3:1-2 in some Bibles)
"I will put my Spirit in you and move you to follow my decrees." — Ezekiel 36:27
"This is the covenant... I will put my law in their minds and write it on their hearts." — Jeremiah 31:33
Now watch how carefully the Bible marks the turning point. When Jesus is baptized, John the Baptist sees the Spirit come down — and this time He does not leave:
"I saw the Spirit come down from heaven as a dove and remain on him." — John 1:32
Remain. That is the whole difference between the old covenant and the new. And here is why it could happen: the righteousness Jesus won on the cross makes a person clean enough for God to dwell in, not just visit. In the Old Testament the Spirit could rest on a man for a while. After the cross, the Spirit can live in you and stay — because you have been made righteous enough to be His temple. That is not a small upgrade. That is Eden again — a living, permanent relationship with God on the inside.
One more thing from Jesus' own life, because He is the pattern. Notice that Jesus performed no miracles until after the Spirit came upon Him. His first sign, the water into wine (John 2:1-11), comes after the dove, not before. And before He left, He told His disciples to do nothing — nothing — until they received the same: "Wait for the gift my Father promised... you will receive power when the Holy Spirit comes on you" (Acts 1:4, 8).
If Jesus Himself waited for the Spirit before He moved, that tells you how essential this is.
In short: In the old covenant the Spirit came and went, resting on a few. The prophets promised a day when God would pour Him out on everyone, to live inside them and stay. The cross made it possible, and Jesus is the first man the Spirit remained on. What was rare became available to all.

