Hi Daniel
Remember to post as much information as you can regarding the people you are researching. Including William's American information- as often more information was recorded at the place of arrival than the point of departure. The more information you post, the more likely it is that one of our volunteers will be able to advise or assist you. Also include information concerning which sources you may have already used so others may further your search.
Some sites that may be of use to you are:
http://www.nationalarchives.gov.uk/records/looking-for-person/default.htm
http://www.irishtimes.com/ancestor/browse/#counties
http://www.nationalarchives.ie/genealogy1/introduction-to-genealogy/
Have you tried checking checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64) for any of Williams' relatives? The 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). 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.
For further assistance you might try contacting:
East Galway Family History Society,
Woodford Heritage Centre,
Woodford,
Loughrea,
County Galway.
Tel: +353 (0)90 9749309
Fax: +353 (0)90 9749546
Email: galwayroots@eircom.net
Galway Family History Society West Ltd,
St Joseph?s Community Centre,
Ashe Road,
Shantalla,
Galway,
County Galway.
Tel: +353 (0)91 860464
Fax: +353 (0)91 860432
Email: galwaywestroots@eircom.net or galwayfshwest@eircom.net
Regards
