Well, I'm more familiar with America's educational system than Ireland's, but fortunately one kind find almost any information on the Web these days. :-)
Here's how I research a paper: First I go to my local library and use their computer to look up my subject. In your case, you would look for "Education." You will probably get a list of books that cover Irish education, since libraries tend to focus on things of direct interest to their patrons. If the field is too broad, narrow it down: "Education in Ireland" or "education 1960s", for instance.
When I've found a book, I look in the table of contents and the index for words that cover my subject. Some will have what you want, some won't. If you have a large library available to you, you can look for educational periodicals. These are most likely to be found in a university library. And be sure to ask a librarian for help. I used to work in a library, and trust me when I say they WANT to help you.
You can also check book stores, of course. I have occasionally bought a used book for research if I thought I might be able to use it more than once, and if the price is right (this means very cheap!).
I supplement the books I find with Internet research
. Just type your subject into a search engine and see what pops up. Again, if you get too many possibilities, narrow down your terms. You can also search for educational journals by name; some will let you download articles for free, but most charge for the privilege.
The very best place to look for academic articles is in the database of a university library, if you have access to one. There you will find articles that you can download from scholarly journals. You search for them by subject, just like with books or on the Web. You are likely to find articles that specifically address the areas you mentioned--state, church, etc. Again, ask a librarian for help.
This is pretty basic researching info; forgive me if you already know it. I took you at your word about not knowing where to start!
I hope this info helps. Post again & let me know how you do!