If you are looking for records dating before 1864, when the state began collecting all civil registration information, your search will take you to church records. Church records are essentially the lone source for family information prior to 1864.
Through the 19th century about 85% of Irish people were affiliated with the Roman Catholic Church, approximately 10% were Presbyterians and 5% were affiliated with the Church of Ireland. Catholic Church records cover vast geographical areas and numbers of people, so these searches can be quite time-consuming. By contrast, Church of Ireland records cover smaller, less-populated parishes, and are comparatively much easier to search in detail.