This is one of those questions that could require a very long or a very short answer. Here we’re going to go with a very short answer.
The Old Testament – God’s spirit
God’s spirit is God’s breath or power. For which the adjective ‘holy’ is usually redundant and not used. The adjective ‘holy’ is found twice with spirit in the Old Testament, but like another subject that is almost absent from the Old Testament (the devil), the holy spirit is not really developed until the New Testament.
The New Testament – introducing the holy spirit.
Wheras ‘holy spirit’, literally ‘holy breath’, is a New Testament idiom or allegory for the breath or activity of God working on men and women, who are by nature ‘unholy’, which is why the ‘holy’ is relevant. The first interactions are with prophets, Mary and Jesus in his baptism.
But otherwise John 7:39 says that the spirit ‘was not’ (probably meaning ‘was not given’) until after the resurrection. So immediately after the resurrection comes one of the the key verses about the holy spirit, when Christ on the day he was raised breathes on the disciples and says “receive the holy spirit” (John 20:22). In the next verse Jesus makes it clear that this is related to forgiving others. The disciples do not receive powers of the holy spirit until 47 days later (Acts 2).
These are more in depth answers:
The Holy Spirit – Bible Understanding of God’s Power by Harry Tennant
The Holy Spirit – Power of Person by Mike Bull
CBM 40 Lesson course – Lesson 20 – God’s Holy Spirit
CBM 18 Lesson course – Lesson 13 – The Holy Spirit
The Holy Spirit and the Believer Today by A.D. Norris