In Acts 2:14–41, Luke records Peter quoting or alluding to a half-dozen passages or so in his Pentecost sermon. These passages are listed below along with a snapshot explanation for why Peter quoted each passage.
Joel 2:28–32a in Acts 2:16–21
Peter quoted Joel to identify the cause for speaking in tongues as something of the same nature of the outpouring of the Spirit that will take place before the end of our present age. As in Joel’s day and as in the future day about which he prophesied, so also it was on the day of Pentecost—all who called out on the name of the Lord would be saved.
Psalm 16:8–11 in Acts 2:25–28
Peter quoted and applied David’s psalm to Jesus. It was not David but his descendent Jesus who had not been abandoned to Hades and seen his flesh corrupted. God raised this Jesus up and exalted Him to His right hand. The proof of this exaltation was Jesus’ having received and sent the Spirit, resulting in many speaking in tongues.
Psalm 132:11 in Acts 2:30
Peter alluded to this psalm, which in turn summarizes the covenant between God and David in 2 Sam 7:12–16 (and sounds much like Ps 89:3–4, 35–37). God swore an oath to David that He would set one of his descendants on his throne forever. This descendant is Jesus.
Psalm 110:1 in Acts 2:34–35
Peter quoted this psalm to identify the exalted position of Jesus at the right hand of the Father. This position was the explanation behind why the people heard Peter and others speaking in tongues. Only Jesus at the Father’s right hand could have been given and poured out the Spirit, which also meant that He was raised from the dead in order to be there.
Isaiah 57:19, Joel 2:32b, and Others in Acts 2:39b
Peter alluded to Isaiah and maybe other texts and then Joel’s prophecy again to instruct his listeners who could receive the Holy Spirit. The promised Spirit and salvation were for anyone that the Lord calls to Himself (Joel 2:32b), children included (similar to many passages – cf. Gen 9:9; 17:7, 9, 10; 28:14; Deut 30:19), even if they were from far off (Isa 57:19; cf. Acts 2:9–11).
Deut 32:5 and Ps 78:8 in Acts 2:40b
Peter exhorted his listeners to be saved from this crooked generation. In the Septuagint, “crooked” is used by both Moses and Asaph to describe Israel in Moses’ day as well, a fitting parallel for Peter’s generation who nailed Jesus to the cross.