Question: "What does the Bible say about infant baptism / paedobaptism?"
Answer: There is much confusion about baptism in the various Christian denominations. However, this is not a result of the Bible presenting a confusing message on baptism. The Bible is abundantly clear of what baptism is, who it is for, and what it accomplishes. In the Bible, only believers who had placed their faith in Christ were baptized - as a public testimony of their faith and identification with Him (Acts 2:38; Romans 6:3-4). Water baptism by immersion is a step of obedience after faith in Christ. It is a proclamation of faith in Christ, a statement of submission to Him, and an identification with His death, burial, and resurrection.
With this in view, infant baptism is not a Biblical practice. An infant cannot place his or her faith in Christ. An infant cannot make a conscious decision to obey Christ. An infant cannot understand what water baptism symbolizes. The Bible does not record any infants being baptized. Infant baptism is the origin of the sprinkling and pouring methods of baptism - as it is unwise and unsafe to immerse an infant under water. Even the method of infant baptism fails to agree with the Bible. How does pouring or sprinkling illustrate the death, burial, and resurrection of Jesus Christ?
Many Christians who practice infant baptism do so because they understand infant baptism as the new covenant equivalent of circumcision. In this view, just as circumcision joined a Hebrew to the Abrahamic and Mosaic covenants, so baptism joined a person to the New Covenant of salvation through Jesus Christ. This view is unbiblical. The New Testament nowhere describes baptism as the New Covenant replacement for Old Covenant circumcision. The New Testament nowhere describes baptism as a sign of the New Covenant. It is faith in Jesus Christ that enables a person to enjoy the blessings of the New Covenant (1 Corinthians 11:25; 2 Corinthians 3:6; Hebrews 9:15).
Baptism does not save a person. It does not matter if you were baptized by immersion, pouring, or sprinkling - if you have not first trusted in Christ for salvation, baptism (no matter the method) is meaningless and useless. Water baptism by immersion is a step of obedience to be done after salvation as a public profession of faith in Christ and identification with Him. Infant baptism does not fit the Biblical definition of baptism or the Biblical method of baptism. If Christian parents wish to dedicate their child to Christ, then a baby dedication service is entirely appropriate. However, even if infants are dedicated to the Lord, when they grow up they will still have to make a personal decision to believe in Jesus Christ in order to be saved.