Statutory records of births, deaths & marriages started in Ireland on 1.1.1864 (1st April 1845 for Protestant and Non-Conformist marriages). Prior to that you are heavily dependant on church records, where they exist. Most early Irish censuses have been destroyed so no help there.
To use church records effectively, because there’s no centralised index, ideally you need to know the ancestor’s exact denomination, townland and parish in Ireland. (Otherwise it’s a needle in a haystack). Good sources are poor law applications which usually contain details of the person's parish in Ireland (Mitchell Library in Glasgow for the Glasgow area). Check every possible census. Census returns sometimes include exact place of birth, instead of just “Ireland”. Look out also for other Irish people nearby in the censuses. Immigrants often settled near people they knew from home. Check any military records, old diaries, gravestones, family bibles, wills and obituaries.
Some Irish parish records are searchable on line, but not every denomination and every parish has agreed to have their records copied and so these websites do not cover Ireland comprehensively.