Hi Kenneth
Thank you for your query. The key to continuing your research is to find an place name of origin for your family in Cork. The first place we recommend that you search is the emigration records as generally more information was given at the port of arrival than the port of departure.
http://www.castlegarden.org/ (1820-1892)
http://www.ellisisland.org/ ( 1892-1924)
http://www.archives.gov/
You can also search Griffiths Valuation or the Tithe Applotment books. These were a nation wide valuation of land ownership and rental in Ireland and can give basic information where church records may not exist.
http://www.askaboutireland.ie/griffith-valuation/
http://titheapplotmentbooks.nationalarchives.ie/search/tab/home.jsp
Civil registration records are available from the General Register Office (GRO). These start from 1864. You can access the website here: http://www.groireland.ie/research.htm
Hopefully you will find a place of origin for your family and you can then search the church records. The link below will allow you to search what Roman Catholic records exist for each parish and the dates etc.
http://www.irishtimes.com/ancestor/browse/ (Browse Roman Catholic Parish Map)
You can also search the distribution of various Surnames (Flaherty, Layton, Reardon, Riordan?) during the mid 19th century on this site.
Church of Ireland parish registers for the period up to 1870-are public records. Registers are available for about one third of the parishes, however many were destroyed in the Public Records Office in Dublin in 1922. Most are still held by the local clergy, although some are in the National Archives of Ireland and others are in the Representative Church Body Library in Dublin. A list of all surviving registers is available in the National Archives. http://ireland.anglican.org/about/42 and http://www.nationalarchives.ie/. The Anglican Record Project is has created an index to their records: http://ireland.anglican.org/cmsfiles/pdf/AboutUs/library/AngRecord/bunclodyunionindex.pdf
Presbyterian registers are held in three main locations: in local custody, in the Public Records of Northern Ireland (PRONI) http://www.proni.gov.uk/index/family_history.htm and at the Presbyterian Historical Society http://www.presbyterianhistoryireland.com in Belfast.
Here are some other websites that may be of use to you:
http://www.nli.ie/
http://www.nationalarchives.ie/
http://www.rootsireland.ie/
http://www.corkpastandpresent.ie/genealogy/
http://mallowheritagecentre.com/
http://www.skibbheritage.com/
I hope that this information is of help to you Kenneth. Do not hesitate to contact us again.
Kind Regards
Genealogy Support