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Peter denied three times before Christ died on the cross for our sins and before Peter received the Holy Spirit.

So it was not a sin?

I don't understand where the Holy Spirit comes in, after all one of the Holy Trinity was present with Peter every day.

Baptism of the Holy Spirit: A few referenced here...

Matthew 3:11: "...He will baptize you with the Holy Spirit..."
Mark 1:8: "...He will baptize you with the Holy Spirit..."
Luke 3:16: "He will baptize you with the Holy Spirit..."
Luke 24:49: "...stay here in the city until you have been clothed with power from on high." (see fulfillment in Acts 2).
John 1:33: "...the one who baptizes with the Holy Spirit."
Acts 1:4-5: "...the Promise of the Father..."; "...you will be baptized with the Holy Sprit..."
Acts 2:1-4: "All of them were filled with the Holy Spirit and began to speak in other languages..."
Acts 2:14-18: "...I will pour out my Spirit..." (quoting Joel 2:28 and 29).
Acts 4:31: "...they were all filled with the Holy Spirit..."
Acts 8:14-17: ...prayed for them that they might receive the Holy Spirit..."; "...as yet the Spirit had not yet come upon any of them..."; "...they received the Holy Spirit..."; "...the Spirit was given through the laying on of the apostles' hands...".
Acts 9:17: "...Jesus...has sent me...that you may...be filled with the Holy Spirit."
Acts 10:44-48: "The Holy Spirit fell upon all who heard the word..."; "...the gift of the Holy Spirit had been poured out..."; "...people who have received the Holy Spirit..."
Acts 11:15-16: "...the Holy Spirit fell upon them..."; "...you will be baptized with the Holy Spirit..."
Acts 19:1-6: "Did you receive the Holy Spirit...?"; "...the Holy Spirit came upon them, and they spoke in tongues and prophesied..."