Now you have your town (or you've always had your town). Two choices now exist, depending on the year you are looking for. But to make the correct choice, you need to know how it works overseas.
What you have been in search of is your town or freguesia. The freguesia belongs to the council (like our counties), and councils belong to a district (a way of dividing the councils for efficient handling of records). The Azores have 3 districts (Ponta Delgada, Angra, and Horta), Madeira has only one (Funchal), and Portugal has too many to list. You will write (if necessary) to the Civil Registry or Biblioteca of a particular district, as described below.
Records are housed 2 main ways. The last 100 years or so are housed in the various Civil Registries. Records older than that are housed in the Bibliotecas e Arquivos. The Genealogical Society of Utah (GSU, today they are called FamilySearch), who did the filming for the LDS and their FHCs, did filming of the Azorean records around 1986, and the Cape Verdean ones around 2004. You can therefore expect that the Azorean film should include up to approximately 1886, but I have seen films that end anywhere from 1869-1883. (The Madeiran records were filmed up to approximately 1910). If you are looking for a birth in within the past 100 years, you will need to write to the Civil Registry. If you are looking for a record from about 1910 or older, they can be found on the Centro de Conhecimento dos Acores (CCA - Center for Knowledge of the Azores) Web site. Madeira's are at the Archive Regional of Madeira (ARM). Access both of these through tombo.pt.
1. Civil Registries - Civil registry began in 1832, although it wasn't mandatory. It began officially in 1878, but mainly used by non-Catholics. In 1911, Civil Registration became mandatory. You must know the municipality/council (like the county) your ancestor came from to use the Civil Registries. Here you will find births, marriages, and deaths. The records vary somewhat from registry to registry; some records have cross-references to other events. The Azores have 19 Civil Registries; Madeira has 10; and in Portugal there are many (Lisbon has 11 all by itself!) (See elsewhere: Geographical Resources for Researching in (Azores/Madeira or Portugal) for the appropriate address). In order for you to identify which Registry to write to, you will need to consult a map. Maps can be found on the Azores GenWeb (Internet search: azores genweb). The fees for records will continue to increase over time. They do take checks or money orders from your bank in U.S. dollars). The fee covers the cost of the certificate, the conversion to escudos and money for postage. A form letter is found elsewhere on the Web.
2. Biblioteca e Arquivo - If you are caught between the CCA or ARM and the last 100 years, you will have to send the form letter to the Biblioteca instead. The Azores has 3 Bibliotecas, Madeira has only one and Portugal has 18. Again, refer to the links found on the Web for the address.
3. The Film - Once you have your records from the appropriate agency, or were lucky enough to start with film, you will find most of your documents on film and available for order from your local FHC. Although, the GSU tried to film everything that was available, things were missed. Some of the oldest books may not have been filmed due to their deteriorating conditions, or they may not be at the Bibliotecas. It appears that for the Azores, mainly baptisms, marriages and deaths were filmed. In Madeira, "passaporte" (emigration records) were filmed. To find the complete listing of what is available in your area of research, you will have to look at the "Topic List" from the FHL Catalog for your locality (Azores, Madeira, Portugal). Once you get into these records, you will find some of the handwritten entries easy to read, some hard to read, others faded, some full of worm holes, etc. If you are having a difficult time reading the film, you may want to note the name and the date and write to the Biblioteca for a clearer copy until you are comfortable reading them. Many of these records were written on a bluish linen-type paper with a brownish, iron-based ink, which either faded or bleed through the paper.
4. Tombo.pt - Since both the Azorean and Madeiran governments have put many vital records, emigration out of the country, various indices online, you will want to use the Tombo.pt site as your gateway into the CCA or ARM. Tombo will display what records exists for your freguesia. It does not cover any books or unpublished manuscripts. You will have to navigate the CCA or ARM directly to find those.