@jonesdn2020
That is true! As an Irish citizen, you will have an EU passport and things are a lot easier.
We are in Madeira and are going through the process of obtaining residency right now. From what we have experienced (using our Irish passports):
1. You will be required to present your passport, NIF, and proof of residency (a lease, in Portuguese, for example) in order to obtain a certificate of residency from the Junta/City Hall- this cost 3 Euros each. I just received this the other day.
2. I was told that I then needed to go to the SEF office (also located at the city hall), present them with the residency certificate (and other documents like passports, NIFs, possibly proof of insurance) and 15 Euros each to process the residency permit.
3. We were told the process will take about two weeks and that we would be notified to return when the permits were ready.
I'll update this after we get through the process tomorrow.
Cheers