The best way to go around searching for your ancestors is through the paper trail. You are looking for birth, marriage and death certificates of your parents and their parents until you can trace it back to the first person from Ireland. Before the civil registration in 1864, these documents were held in the individual parishes.
There are a few handy websites you can look at to help out with your search like Roots Ireland
, Family Search
and Ireland Reaching Out
Hope this helps and good luck!