In order to look for birth records you'll need to know:
The approx year of birth
The parents first and last name
The town and parish that the family was from (I was able to get the parish info from the ship registry)
If the records are more recent then you'll find them at the Conservatoria do registo civil. If you are standing in front of the City Gates facing the water the building is on your right. The office is on the second floor.
If the records are older than 100 years then they will be at the libray in Ponta Delgada. The library is two or three blocks away from the Conservatoria do registo civil. The records are on the 2nd or 3rd floor. You'll need to fill out a slip of paper to request the year, parish and type of record that you want to look at (marriage or baptismal). Some of the librarians were more helpful than others and only one or two spoke English.
Since the records are in Portuguese I found it helpful to write down the translations for numbers, months and days so I didn't have to keep flipping through my dictionary.
Names will need to be translated too ex. John/Joao
The records are actually babtismal records so even though you know Grandma was born in January she might not have been baptized until March. So, you'll find her in March.
The first name of the baby is listed on the left side of the page and then with in the paragraph you'll have to search for the date and parent's names.
Once you find the record you can ask the librarian to make a photocopy. I think it cost a 1 or 2 Euros per copy. The first day I was at the libray all afternoon and then at closing tried to get the copies I wanted. They wouldn't make the copies because they were closing so I had to go back on Monday. Thank goodness my flight was on Tuesday! I would suggest getting the copies made as you find them just to be safe.
If you have any other questions let me know.