You not only need to know the specific county of origin for your family but also their parents names, the town/townland, civil parish of birth in their county in Ireland and religious denomination.
There are no census records for the 1800s and civil registration didn't start until 1864 (Protestant marriages in 1845). Church registers for the specific town/townland, civil parish of birth are the source for baptismal information. The registers aren't centralised or computerised and their availability as well as how they can be accessed depends on the location as well as the religious denomination.
There are over 2,000 towns/townlands in County Roscommon and nearly 1400 in County Sligo. Here's a map showing where the counties are:http://philnorf.tripod.com/ireland.htm
So the family's civil and church records in the US would be your source to find the information you need before you begin researching in Ireland.
The emigrants' civil and church marriage and death records, obituaries, cemetery records, tombstones, wills and US naturalisation records are all sources for research.
Their children's civil and church birth, baptism, marriage and death records may include their parents place of birth in Ireland.