There are Roman Catholic church records available for the diocese of Annaghdown (Tuam). These date from 1834 for both baptismal and marriage records. You can find these at the National Library of Ireland (NLI), Pos. 4219, or the Church of Latter Day Saints, 1279206, item 1-2. Or the Galway Family History Society West (http://galwaywest.rootsireland.ie/). If you have any difficulty, you could try writing to the parish priest for more assistance.
Civil registration records are available from the General Register Office (GRO). These start from 1864 however. You can access the website here: http://www.groireland.ie/research.htm
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64). Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at. Tithe Applotment Books (1823-38). Microfilm copies of the books for all of Ireland are available at the National Archives of Ireland (NAI) http://www.nationalarchives.ie/genealogy1/genealogy-records/tithe-applotment-books-and-the-primary-griffith-valuation/ or the Church of Latter Day Saints (LDS).
There are estate papers for the area too. Landlord Francis Blake Knox: Rental, 1845-66. National Library of Ireland (NLI), Ms. 3077.
Do you know much about their emigration? Dates, the reason why they left, etc.? Generally, more information was given at the port of arrival rather than the port of departure. If you knew which city they arrived at, this could be a good place to find more information, and perhaps even find out an exact place of origin. There are lots of books and documents available about the Irish emigration to NZ. There is a website you could look at: http://www.teara.govt.nz/en/irish/2 for information about the Irish in New Zealand.
Please make sure you link anyone else in your family who is interested in their Irish heritage to our site - and indeed anyone else you know of Irish heritage.
Genealogist (Ireland XO)