Jean,
Statutory recording of births and RC marriages didn’t start in Ireland till 1864. Prior to that you are mostly reliant on parish records. They generally don’t have as much information in them as appears on the civil certs. You just have to try and work out possible connections from what is available. There is no easy way. So if the townland (address) is given in the church record, does that tie in with what you know of the family? Likewise if an occupation is mentioned. When they marry later (post 1864), does the father’s name and occupation match with what you are expecting? See who is in the townland in the 1901 census. Does that fit with what you were expecting. For example, if there’s a Patrick in the townland and another one who emigrated, then it probably isn’t the right family, or at least not the right branch.
Prior to Griffiths, if the family are in it, you can look up the tithe applotment records (1820s/1830s) but for the average Irish family that’s probably about as far as you’ll get.
The Griffiths Revaluation records followed the initial valuation (ie the one that is on-line). The revals are not on-line. For Mayo, they are in the Valuation Office in Dublin, and will show when a property changed tenant/ownership, and so you can see when someone died etc from that).
You can check for wills (National Archives Dublin), if you think they had sufficient property to merit a will. Most pre 1900 have been destroyed (in the 1922 fire) but some still exist. And the indexes to the wills exist which can point to dates of death etc. See:
http://www.nationalarchives.ie/genealogy1/genealogy-records/...Also check gravestones in the local graveyards. (Not all research can be done on-line).
Elwyn