It starts with the presupposition that God deals with his people by way of covenants. He is a covenant-making and covenant-keeping God. Most Christians will agree that God made a covenant with Adam, a covenant with Noah and a covenant with Abraham. Sometimes when God makes a covenant, He also makes a sign to signify the event. The rainbow serves as a sign of God's covenant with Noah and circumcision was a sign of God's covenant with Abraham. God frequently states that his covenant promises are "to you, and to your children" or "to you and to your generations." When God makes a covenant with a single person, it doesn't mean that only the one person is under the covenant. In the case of God's covenant with Abraham, God specified that all of the males born into Abraham's family were to receive the sign of the covenant in their flesh.
I don't know of any Christians who believe that everyone who received the sign of circumcision was necessarily a believer, elect, or saved. The sign isn't a proof of God's love, but it does signify who is to treated as part of the visible covenant community. God said that those men who did not have the sign of the covenant were to be cut off from their people.
Baptism is the sign of the New Covenant. In Colossians 2:11-12a we read
In whom also you were circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ: Buried with him in baptism...
The sign is given to all those who profess Christ and to all in the household of a believing man, including children, servants, relatives, etc. just as circumcision was done to all males of the house under the Old Covenant.
To recap: God deals with His people through covenants. God commands us to remember these covenants by way of signs. God made a New Covenant in the blood of Christ and the sign of that covenant is baptism. The sign of the covenant is given to all professing believers and to those under their household authority.
Our duty to our children is found in the great commission. Preach the gospel, baptize them and make disciples, teaching them to obey all that Christ commands.