You are getting some good advice. Once you get into your family history (genealogy), you may want to start spending money in the RIGHT places. But you have the right idea to start.
You should get everything you can from your parents first. Interview them for valuable clues. Here are some sites I like, sorry for repeating sites that others are giving.
Use GENFORUM to find your family SURNAMEs genealogy fourms, to talk to people doing similar research.
When you go to Cindy's list, do a search on IRISH. That will give you a filter on Irish sites.
Use puplic libraries in a town where you know someone died! Write to the librarian, or call them, and request an Obituary - that will be full of free information.
LDS FAMILYSEARCH.COM is really good information in all countries for your ancestors (not living). They have census, death, obit, and pedigree files. All free.