FAQ (I am not a lawyer and all the information below is based on my personal experience and understanding; and may not be correct in all cases. Laws and procedures in Portugal are notoriously unclear.):
How can I change my address? You have to go to the local city administration office (loja do cidadão) with your new lease and change it there. The post office will not do it for you.
What are my medical care options in Portugal? If you are a resident, you can register with your local public health clinic (centro de saúde) and are given a number. This will allow you to have access for the VERY reasonable public health system (an American friend had a complete cancer operation with care both before and after, including all specialists for less than 100 Euros in the public hospital system). However, there are reasons why a robust private health system exists like the CUF network of hospitals. This is more expensive but still VERY reasonable compared to insured costs in the US (god help you if you are not insured in the US). But to lower this cost, there are a number of private medical insurance options. I chose one from the Auto Club of Portugal (ACP) which for a total of about 100 Euros a year gets me a card which cuts in half the cost of private medical care, covers the cost of hospitalization, and for another 15 Euros or so, eliminates the 2000 Euro deductible for hospitalization. Did I mention it costs only about 100 Euros? I saved that on my first of many visits to the CUF doctor for an injury.
If you are applying for residency, either visa or permit, they will ask you to show proof of medical insurance that will cover you in Portugal. There are many options on-line. I chose the World Nomad worldwide coverage for about 1000 Euros a year. Both the Consulate in the US and SEF (immigration office) in Portugal will ask for proof. Better have it.
So, why does a private health system exist? It can take a few months to have something NON-urgent taken care of compared to a few weeks in the private health system. You may have to wait a few hours in the public system to see a doctor compared to 5 to 20 minutes to see one in the private system. The same doctors work in both the private and public systems. So, the care is the same.