Now that I am naturalised, I reckon I should share what a first time applicant whose recently got naturalised needs to do to get a passport.
The application forms can be got from any garda station or postoffice (well most post offices have them). As I had previously mentioned, you will need 4 photographs (two of which need to be signed by your local garda). An original copy of your birth certiticate, a marriage certificate if you are married, your naturalisation certificate, proof of residence (a utility bill of some sort) and existing id to verify you are you.
Make sure you put your name on the passport application whilst making sure it is the same as what is on your naturalisation certificate. Make sure it matches and if like me if your name on your birth certificate is different it is worth ticking the box which makes a note of your birth name in your passport.
It takes about 20works days if you apply at the passport office like I did. I chose to do it at the passport office as I had questions to ask to make sure my application was correct. In hindsight I would have applied via the postal passport express service.
Something that I did do was make a photocopy of the naturalisation certificate and went to the local garda station to put a stamp and signature on the document to state that it is a verified copy. This was just in case if I lost the naturalisation certificate for whatever reason.
You can track the application at http://www.dfa.ie/home/index.aspx?id=2859, you'll get a code from the application so don't worry.
24 months and two weeks