Hi Dianne,
To begin with you need a lot more information.
B,D,& M certificates would help find parents names and places of birth, etc.For example the 1878 birth cert. of your Grand Mother should tell
you her parents names & where they came from.
The "Shanleys" seem to all have come from County Leitrim, but prior to 1835 there are almost no records. With the knowledge of her parents names you should then check the shipping records & or the Ellis Island records of their entry into the USA.
Hope this helps.......Good luck