Sacramental records (information about marriages, baptisms, confirmations, first communions, etc.) generally stay in the parish that generated them. If the parish no longer exists, such records should have been sent to the diocesan archives of the local church. Records do not always survive, and not every pastor (not every bishop) recognizes the importance of preserving them. But records that do survive are probably in the files of a parish, diocese, or archdiocese.
Many public libraries acquire the Official Catholic Directory every year. This volume lists all of the dioceses and parishes in the United States and includes such details as names, addresses, and phone numbers of pastors and diocesan archivists, geographical extent of dioceses, and date of their creation.
This last item is important because the diocese to which a parish belongs can change over time. For example, Catholics in Northern Indiana before 1808 would have been served by travelling missionary priests under the authority of the bishop of Baltimore; from 1808 until 1821 they would have belonged to the Diocese of Bardstown; from 1821 until 1834 to Cincinnati; from 1834 to 1857 to Vincennes; and after 1857 to Fort Wayne.