I don't have any experience of searching in the US, but others have who might be able to help on where to look. However when I have some basic information I put it onto Ancestry in the form of a basic tree, this will include DOB and estinated marriage date and estimated death details, country of residence and birth if known. Make some basic assumptions on ages most people get married for instance when they are 18-21, the first born normally follows on within two years, and in your case you have information on names of children, maybe the order of birth. Ancestry will help by looking up information and other trees of people with similar family details. If your lucky someone else will know something about your family members, or they form part of their tree if the married into other families.
Its worth a try even if you only take a minimum subscription. When you get some firm dates try looking in the LDS indexs for Ireland, and the general registration indexes are also available for 1865 onwards in most cases. There are also several other places based on location and parishes some or free to search others you pay as you search. LOcal historical Societies can be very helpfull too. You seem to have your starting place in Cork which is something many peiole don't get that much.