Hi Tina,
There are Roman Catholic church records for the diocese of Aughagower (Tuam). These date from 1828 for baptismal records and 1854 for marriage records. You can find these at the National Library of Ireland, Pos. 4210, or the Church of Latter Day Saints, ref. no. 1279209 item 19-20; 1279210, 1-3. 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
Do you know anything about their emigration? Have you tried checking the Ellis Island records (www.ellisisland.org) or the Castle Garden records (www.castlegarden.org)? 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.
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffth'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.
If they were still living in Ireland before they emigrated, then you might be able to find them on the 1901 or 1911 census here: http://www.census.nationalarchives.ie/search/
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.
Kind regards,
Sinead Cooney
Genealogist (Ireland XO)