Several baptisms are mentioned in the New Testament, typically:

• John's baptism in water for the remission of sins
• Baptism in the holy Spirit by the Lord Jesus
• Baptism in water into the name of Jesus Christ.

There are those who claim that baptism in the holy Spirit is still current yet they also practice baptism in water. But when Paul wrote to the Ephesian Christians he stated there is one baptism.

Ephesians 4:3-6 Endeavouring to keep the unity of the Spirit in the bond of peace. There is one body, and one Spirit, even as you are called in one hope of your calling; One Lord, one faith, one baptism, One God and Father of all, who is above all, and through all, and in you all

This tells us that just as there is only one Lord, one faith and one God and Father of all, and equally so, there is only one baptism. Evidently by the time Paul wrote this letter there was only one baptism and other baptisms had been phased out.

Which was that one baptism at that time Paul wrote this letter?
First we need to understand that in the Bible the word "baptism" means: immersion (Romans 6:1-5 Colossians 2:12) for example immersion in water, or figuratively: an overwhelming, such as when the twelve Apostles where overwhelmed with the holy Spirit on the day of Pentecost (Acts 1:26-2:14).

John's baptism in water had been superseded because in Acts 19:5 those who were immersed into John's baptism were re-immersed, this time into the name of Jesus (Acts 19:5). However the apostles and the early church never ceased practising baptism in water. This then was the one “baptism.” That it could not be baptism in the holy Spirit will be seen in the passages of scripture quoted below where there is an asterisk *.

Who should be baptised in water?

All who hear the Gospel call.

Acts 2:39 For the promise is to you, and to your children, and to all who are afar off, even as many as the Lord our God will call.

AND believe the Gospel

Mark 16:15-16 And he (Jesus) said to them, Go into all the world, and preach the gospel (the good information) to all the creation. He who believes and is baptised will be saved; but he who disbelieves will be condemned.

AND repent.

Acts 2:38 Repent, and be baptised every one of you in the name of Jesus Christ for the forgiveness of sins

[Repentance is not just being sorry. It is the decision to change EDITOR]

Believers in the Lord Jesus have to repent before being baptised for their sins to be forgiven. Like Saul of Tarsus, at baptism they are appealing to the  Lord for their sins to be forgiven.

Acts 22:16 and now why are you waiting, get up and be baptised and wash your sins away appealing on the Lord's name.

So Peter calls it a good conscience's plea to God based on the fact that Jesus rose from the dead.

1 Peter 3:21 ... now baptism also saves us, not by removing dirt from the flesh, but a good conscience's appeal to God on account of Jesus Christ's resurrection. [compare with 1.Peter 1:3]

The message of the Gospel is God's way of calling us to turn to Him.

2 Thessalonians 2:14 Unto which he called you by our gospel, to obtaining the glory of our Lord Jesus Christ.

This applies to folk of all nations, both Jews and non-Jews until the end of this age.
Before He ascended to heaven the Lord Jesus gave his parting instruction to the Apostles to baptise those who implicitly believe in Him:

Matthew 28:18-20 And Jesus came and spoke to them, saying, All authority is given to me in heaven and in earth. Therefore you go and make disciples of all nations, baptising them into the name of the Father, and of the Son, and of the Holy Spirit, teaching them to observe all things whatever I have commanded you: and, lo, I am with you always, even to the end of the age. Amen.

[* Note: This is baptism in water and not baptism in the holy Spirit, because baptism in water was to be done by the Apostles whereas baptism the holy Spirit was done only by the Lord Jesus (Matthew 3:11) SEE FOOTNOTES]

Why baptise people in water? What does it signify?

When repentant believers are baptised in water correctly, they enter into a close relationship with the Father, the Son and the holy Spirit

Matthew 28:19 baptising them into (Greek: eis) the name of the Father, and of the Son, and of the Holy Spirit

It is done for repentant belivers' sins to be forgiven.
Several days after the Lord Jesus ascended to heaven the Apostle Peter carried out the Lord's parting instructions. He commanded sorrowing believers to:

Acts 2:38 Repent, and be baptised every one of you in the name of Jesus Christ for the forgiveness of sins, and you will receive the gift of the Holy Ghost.

The baptised person will receive the indwelling of the holy Spirit in their lives. [SEE FOOTNOTES]

They become members of the Lord Jesus' church.
The Lord Jesus adds repentant believers to His church when they are baptised for the forgiveness of their sins. (cp. Acts 2:40-41)

(Acts 2:47) the Lord added to the church daily those who were being saved.

Acts 2:40-41 And he testified and exhorted with many other words , saying, Save yourselves from this perverse generation. Then those who gladly received his word were baptised: and the same day there were added to them about three thousand souls.

It is both is burial and resurrection.
The repentant believer is buried with the Lord Jesus in baptism. They bury their old life. They rise with Jesus. They rise from baptism to a new life.
Reminding Roman and Colossian Christians how they became Christians Paul wrote:

Romans 6:3-4 Do you not know that as many of us as were baptised into Jesus Christ were baptised into His death? Therefore we are buried with Him by baptism into death, that just as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life.
Colossians 2:12 ... buried with Him in baptism, in which also you were risen with Him through the faith of the operation of God, who raised Him out of the dead.

[* Note: This is baptism in water and not baptism in the holy Spirit, because the baptism in Romans 6:1-5 and Colossians 2:12 is both a burial and a resurrection. The Apostles were overwhelmed by the holy Spirit but they were not raised from out of the holy Spirit. Baptism in the holy Spirit typifies immersion in the sense of the person being overwhelmed but it does not typify resurrection, so the Baptism in Romans 6:1-5 and Colossians 2:12 cannot be the baptism in the holy Spirit. SEE FOOTNOTES]

They all become children of God and put on Christ.

In the one baptism, that is baptism in water, the repentant believer "puts on Christ"

Galatians 3:26-27 For you are all the children of God by faith in Christ Jesus, because as many of you as have been baptised into Christ have put on Christ.

There is one faith.

Some say there was one way of salvation for the Jews [salvation by baptism in water for the forgiveness of sins] and another way for the non-Jews [salvation before baptism in water].
Paul reminded the Galatian Jew and non-Jew Christians, that both Jew and non-Jew become Christians the same way:

Galatians 3:28-29 There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for you are all one in Christ Jesus. And if you be Christ's, then you are Abraham's seed, and heirs according to the promise.

We have to insist on adherence to that one faith which was given at the beginning.

Jude 1:3 Beloved, when I gave all diligence to write to you of the common salvation, it was needful for me to write to you, and exhort you that you should earnestly contend for the faith which was once delivered to the saints.

Ephesians 4:3-6 Endeavouring to keep the unity of the Spirit in the bond of peace. There is one body, and one Spirit, even as you are called in one hope of your calling; One Lord, one faith, one baptism, One God and Father of all, who is above all, and through all, and in you all.

This tells us that just as there is only one Lord, one faith and one God and Father of all, and equally so, there is only one faith. That one faith was given just once. We must contend for it and stick to it.

"If it is new - it cannot be true"

Some admit that initially baptism in water was for sins to be forgiven but claim that was only at the beginning. They claim the way of salvation is now different.

Those who introduce or teach any other way of salvation are to be rejected.

Galatians 1:8-9 but even if we (i.e. the apostles) or an angel from heaven, preaches any other gospel to you contrary to what we declared unto you, let him be rejected. As we have said before, so I also say again, If anyone preaches any other gospel to you contrary to what you received, let him be rejected.