Is a baby saved initially through baptism and the faith of his parents, and when must he have his own personal faith?

Yes, you are saved initially at the sacrament of baptism, because that is when you receive sanctifying grace that justifies you before God. Your soul is changed substantively from a dead soul to a living soul that can now take in the grace of God, and you grow in that grace throughout life. This is what Jesus tells Nicodemus in John 3:5, that unless a man is born of water and the Spirit he cannot enter the kingdom of God, and the Council of Trent uses that verse for baptismal justification. Most Catholics are baptized as infants, and if a baptized infant died it would go straight to heaven with a pure soul. We do not see any visible change at an infant's baptism, no lights or singing; the change is invisible and known by faith, unlike the dramatic conversion experiences some Protestants tie to being born again. As for when the child needs his own personal faith: as soon as he can. That is the whole point of the age of reason, around seven years, when he becomes accountable, when he can premeditate and understand that a wrong act is a sin before God requiring repentance. Before that, small children act on impulse, not premeditation, so they are not treated as culpable. From about seven onward he can no longer depend on his parents' faith; he must absorb what they taught him and have his own faith in Christ. Logically it could be five, six, seven, eight, or nine, but the Church has chosen seven, and we respect her authority.