There is an excellent discussion of baptism going on at Bunnie Diehl blog. The conversation starts by asking for an explanation of why some churches will not recognize a person's previous baptism. It predictably turns to a discussion of whether a baptism is merely an outward symbol of a person's born again faith or whether it is God reaching out to the person through the water, as He said he does, to create faith in the heart of all...even of an infant. There are many more issues and related doctrinal matters, such as a person's inability to nominate themselves for salvation, whether a person can lose faith if it is not sustained by the Word and whether it is important to surround yourself and your family with scripturally sound teaching.
Getting back to the discussion on Bunnie Diehl, I particularly I loved Andrew's comments:
Is it not significant that the "all nations" Jesus commands His disciples to baptize in Matt 28:18ff has been building up through the whole Gospel of Matthew to speak of little children, infants, not-so-bright people, Gentiles, ladies, and lepers?
11:25 "I praise You, Father that You have hidden these things from the wise and intelligent and have revealed them to infants."
18:6 "these little ones who believe in Me", implying that infants can have faith
18:10 "See that you do not despise one of these little ones, for I say to you that their angels in heaven continually see the face of My Father who is in heaven."
19:13-15 "Let the little children [infants in Luke] come to me, and do not hinder them..."
21:15 When the chief priests question the children's praise of Jesus, He responds, "Out of the mouths of infants and nursing babes You have prepared praise for Yourself."
The consider what the Apostles will do following those words...
Polycarp, born AD 69, a disciple of John, is going to be baptized as an infant.
Justin Martyr (AD 100-165) is going to write about people who were Christ's disciples "from childhood".
Irenaeus (AD 130-200) is going to write that infants are being baptized in the churches.
Origen (AD 185-254) is going to write that infant baptism was a practice of the Apostles.
And the list goes on thru the centuries...
Did Matthew and the early church get it wrong?