Hello, Maggie and welcome.
Civil registrations of births, deaths and Catholic marriages began in 1864. Civil registrations of marriages other than in a Catholic church began in 1845. Do you know the religious denomination of the Kinnamore family?
Irish Genealogy.ie is the official government website for finding BMD registrations. https://www.irishgenealogy.ie/en
It's free to use. You simply sign in, tick the box and solve a catchpa.
There was a death registration of a Thomas Kinnamore, aged 75 in Loughrea Superintendent Registrar's District in 1865. The register page is not yet open to view on the site so if you think this person might have been your GGF you either need to order a photocopy or the certificate or wait until images of that register come online. Athenry is in Loughrea district.
Death registration for a Bridget Kinnamore, widow, aged 84, in Loughrea District in 1880. This page of the register is visible. Bridget died at the Union hospital, Loughrea. (Union was the Poor Law Union. The hospital was part of the workhouse.) Informant was a member of workhouse staff. Bridget's age may have been an estimation, as might Thomas' age. Occupation column says "labourer"; I don't know if it meant Bridget was a labourer or that she was widow of a labourer.
There were 2 other Kinnamore entries for death registratons in Loughrea District between 1864-1901. They were both in 1865 and were for Selia and Lelia, both aged 74. Images not viewable. One may be an amended entry, so possibly only 1 person.
I don't see any other births, marriages or deaths for Kinnamore during that period. There may have been alternative spellings. Districts adjacent to Loughrea are Tuam, Mountbellew, Balinasloe, Portumna, Scarriff, Gort and Galway.
Some R.C. parish registers for Athenry are on National Library of Ireland website. https://registers.nli.ie/parishes/0103
It has marriages 1821-9 & 1858-78 and baptisms 1858-78, so too late for baptisms of the people you named. Follow links at foot of the page for possible locations of other registers for the parish. A marriage usually happened in the bride's parish. The map on Athenry parish page shows nearby parishes.
A Mary Kinnamore was baptised in New Inn parish, Diocese of Clonfert, County Galway 9th Sept. 1834. She was illegitimate daughter of a Thomas Kinamore and Mary Hart. A John Kinamore married in the parish in Feb. 1835 to Bridget Healy. New Inn parish was next-but-one parish to Athenry. It's adjacent to Loughrea parish.
Griffiths' Valuation (1850s) has 3 Kinnamore entries for Galway, all called John. Griffiths lists only property holders. www.askaboutireland.ie/griffith-valuation
A useful guide to Irish family history is Irish Genealogy Toolkit http://www.irish-genealogy-toolkit.com