Hi, and welcome!

Not sure what the SEF is, but I take it you're headed to Portugal. Your best bet is to contact the Portuguese Consulate in New Zealand (or wherever you're living now). Try online to see if the Portuguese Consulate (or Embassy) has a website for the country where you now live. They usually have most of the information on their website, or at least enough to get you started.

To stay longer than three months, you need to have your visa (a "long-stay" visa) when you enter the country so the Immigration people can stamp it to validate the day you entered. You generally take that to some authority (varies by country - often the police or local foreigners office) to obtain a residence permit, which functions like a national i.d. card for non-citizens.

The type of residence permit you get (working vs. non-working, student, or other type) is determined by the type of visa you entered the country on - and despite all the fuss made about the Schengen treaty, long-term visas obtained for one European country are not transferable to another EU country. If you move countries, you have to start again with a new long-stay visa.