Gargaro (Connor) family search

Hello my name is Tom Gargaro. My mother's maiden name was Connor. The most information that I have about her ancestors is that her great-grandfather was the first that I can find that was born in Ireland. His name was John.G. Connor and he was born in Ireland in 1839. I know that by 1850 he was living in Pittsburgh, Pa. His father's name was Arthur and was born in 1781 and his mother's name was Elizabeth . I don't have any additional information regarding county, parish, siblings, etc. Any help on how or where I can get started would be greatly appreciated.