Look for birth, marriage and death certificates at the state registry or at the local city and town registries. Look also for baptisms and marriages in the local church or archdiocese archives. A good beginning place is the 1900 or 1920 Federal census. They are indexed and available in the regional National Archives around the country, and also at many larger public libraries. These two census records will give the year of immigration and whether or not the male has been naturalized.
Many Irish names are extremely common.
Even if you think the name Florence is uncommon in America, you will find hundreds of Florence McCarthys from County Cork. Don't just focus on your immediate relatives. Spread out your research. Look for siblings, cousins, friends, and neighbors. The Irish tended to cluster by village in the New World.