To understand baptism, we cannot just look at a few NT passages that mention baptism explicitly. The apostles clearly believed there were events in the OT that prefigured Christian baptism.

For example, Peter says the flood, in which Noah and his family were saved by water in the ark, was a type, or foreshadowing of Christian baptism. Paul says the Red Sea crossing, out of Egypt and into freedom, in which the glory cloud poured out water from above, was a model of Christian baptism. 
The truth is that the OT is full of types of baptism. Those types pointed ahead; they were anticipatory and promissory.
But now Christian baptism announces the fulfillment of those types and shadows. The promises God made in the past are fulfilled in the waters of baptism. This one Christian baptism in which we all share fulfills all old covenant baptisms. Baptism announces the promised kingdom is NOW here — and in baptism we enter that kingdom. This baptism you will witness today is every bit as much God’s work as the flood and the Red Sea crossing. Those were mighty acts in which God rescued his people; baptism is the same kind of thing. This is an act of God’s saving grace and power, an effectual means of salvation.
These are the waters of life, the waters of salvation, because God is present here. Baptism is not merely water, it is not even merely water applied with the Triune formula. Baptism is not merely an act of the pastor or the church. Baptism is an act of God: the Father working through the water to sprinkle us with the blood of the Son and to pour out his Spirit upon us. Every baptism is an opportunity for you to not only witness a mighty act of God. It is also an opportunity for you to remember your own baptism and give God thanks.