What the Gospel IS: Adoption

But when the fullness of time had come, God sent forth his Son, born of woman, born under the law, to redeem those who were under the law, so that we might receive adoption as sons. And because you are sons, God has sent the Spirit of his Son into our hearts, crying, “Abba! Father!”  So you are no longer a slave, but a son, and if a son, then an heir through God.

Galatians 4:4-7

Below is an apt description of the doctrine of Adoption from the Westminster Confession of Faith, Chapter XII

All those that are justified, God vouchsafes, in and for His only Son Jesus Christ, to make partakers of the grace of adoption,[1] by which they are taken into the number, and enjoy the liberties and privileges of the children of God,[2] have His name put upon them,[3] receive the spirit of adoption,[4] have access to the throne of grace with boldness,[5] are enabled to cry, Abba, Father,[6] are pitied,[7] protected,[8] provided for,[9] and chastened by Him as by a Father:[10] yet never cast off,[11] but sealed to the day of redemption;[12] and inherit the promises,[13] as heirs of everlasting salvation.[14] .

Also, in his sermon “Praying in the Spirit”, Tim Keller describes how our relationship to God is changed by adoption through the Spirit, and clarifies the language Paul uses to describe it,

Many of you have heard that the Aramaic word abba is best translated “daddy”…..but that is not completely accurate, it’s best translated “dadda”……an 8 year old doesn’t say this word, it’s an infant. It’s primal, it’s instinctive, it’s the very, very first word out of a kids mouth…..why would Paul use this language? What infants want when they say “dadda” is they just want dadda, they want to grab the neck, get up close to the face, or just see the world as daddy sees it up there…..Paul is saying that when you become a Christian, you get the Spirit, you get a new language…….it’s a desire for God himself, you’re after the nearness itself…the Holy Spirit creates a primal language where you just want to be with God, you have a thirst for Him.