NIF - they haven't given out cards for some years now, they will give you a stamped print-out, the original of which you should keep with your file of key paperwork. This will be accepted - just quoting the number sometimes is, sometimes isn't.
SNS number - recent feedback suggests that many/most/all Centros de Saúde (health centres) are not processing applications, but instructing people to apply via email to an address they give you. This process appears completely stalled - recently arrived friends have waited 2-3 months. However, this IS the default COVID position, unfortunately.
Which Centro de Saúde - theoretically, it should be the one covering your registered address (which is normally the nearest one, though urban zoning can be slightly off, as you might expect). If you end up at the wrong one, they should be able to tell you which one is the right one. However, it's sometimes possible to get registered at another CdS within the same health region (ARS - look for something like ARS Lisboa Central, and then google whatever the relevant one is, to find where its other CdSs are.
Note - if you have a genuine medical emergency, and don't yet have a number, then it should be possible for them, exceptionally, to sort the SNS number when dealing with you, at the CdS, or at the hospital. Please don't use this route unless it is critical!