The baptism referred to here is NOT WaterBaptism though Peter may have thought that was what Jesus was talking about. Jesus clearly stated that those who would be His disciples must ^abide in Him^ and that ^His Words must abide in them^ Joh 15:7. This is the immersion, or baptism, that He expects of all would follow Him Joh 8:31.
Peter, who instructed John Mark on the events of Christ Jesus^ life, was a Jew fully versed in Judaism and purification rites.
Peter assumed that those rites had spiritual efficacy, which they do not. Rites are symbols of the real thing, but they are not the real thing.
15 years after the resurrection and ascension of Christ, Paul had it out with the Judaism Christians in Jerusalem. The Jews reluctantly received Paul^s instruction and enforced only 3 laws on those who come to Christ: -> do not eat things offered to idols -> do not drink blood -> do not fornicate
So, the early believers had very little dogma. They believed that: -> Jesus was the Messiah -> Jesus had raised Himself from the dead and ascended to heaven -> Jesus had given them His HolySpirit
Judaism defined everything else. The Gentiles did not have a history of knowledge of Judaism and so Paul preached to them the pure Gospel of GraceThroughFaith in Christ Jesus our Creator, MightyGod, Savior, Lord and King.
We are to be immersed into Christ Jesus Joh 15:7.
It is not enough just to believe that Jesus is the Christ. We must immerse ourselves in Him Joh 17:21 being BaptizedInTheNameOfJesusChrist.
How do we do that? -> absorb the Word -> SpiritBaptism -> have the MindOfChrist
As to WaterBaptism, it is public evidence of faith in Christ. It is not evidence to the Lord as He knows the heart of man.
Paul makes it clear that SpiritSalvation is a result of HeartFaith in Christ Jesus and oral confession that Jesus is the resurrected Christ in Rom 10:9,10.
Public Confession of Christ confirms our salvation to ourselves and those to whom we witness. It is a precursor to lifelong sharing of the TestimonyOfJesusChrist.