Dear Jane:
Thank you for your post to the Ireland Reaching Out message board and welcome to the site.
I note your attachments and your comments about County Waterford. This may be a good place to start as both surnames (Gooley and Fitzgerald) were there and it would appear that your ancestors may well have known one another before they emigrated.
I have found a Michael Gooley born about 1852 in Cahir Waterford to parents names John Gooley and Honor Shunny. This is from the Roots Ireland Database which is a subscription site that has transcribed records from all over Ireland. This may or may not be your Michael.
In addition, I noted that there is a family tree on Ancestry that is associated with Michael Gooley which gives him a year of birth of 1845 based upon his death certificate of 9 October 1887. It also lists his parents as Martin and Mary. I was not able to find any corresponding birth/baptism record for a Michael Gooley with these parents, but it is possible that the registers are not on this site, and may be held in the parish.
I also noted that Michael Gooley was listed as a blacksmith. Often, that trade tended to run in families, so it might be worth looking further to see if there were other Gooleys who were also trading as blacksmiths. You mentioned a James Gooley, a brother living with Michael and his wife, Kate in 1870. Have you traced him? He may give you further clues as to Martin's parents as well as to the parish that they came from.
I have found a Martin Gooley aged 70 in the 1901 Census living in Portlaw Waterford with his wife Alice.
http://www.census.nationalarchives.ie/pages/1901/Waterford/Portlaw/William_Street/1768503/
You might also take a look at the Griffith's Valuation for Gooley families living in Waterford.
There is a Patrick Gooley living in the townland and the civil parish of Kilbarrymeaden.
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=48&Submit.y=15&familyname=Gooley&firstname=First+Name&baronyname=&countyname=WATERFORD&unionname=&parishname=
If you can gather as much information as possible about your ancestors from the Chicago area to include gravestone information and cemetery records, that would be a big help to you with your research.
If you have any questions, please let me know.
Kind regards,
Jane.