Credit cards which earn frequent flyer points is a popular way to earn frequent flyer points. You can receive thousands of points on everyday spend. And, many of these cards offer generous signup bonuses! Compare to find the credit card that best suits your needs.
If you don't fly NZ much then I would not credit to Airpoints. Did you know there is an annual fee if you do not fly 1 paid Air NZ flight every year? It is possible that your existing account has been closed.
First, if you intend flying Singapore Airlines in first or business class again, even if only modest chance, sign up for Kris Flyer. You can still credit points to another program and they will track your spend on SQ first and business class flights - once you reach S$25,000 in a year you qualify for PPS status which gets some extra benefits on SQ (the most important of which in my opinion is the ability to clear SQ waitlists).
If this is the only *A trip you'll ever do then I'd sign up to a program that maximises the award you can get from this trip.
UA Mileage Plus has cheap awards but low earning rate - you'll earn something like 36k points, enough for award trans-tasman in business class.
BD Diamond Club requires european address (although reports on FT are this is no longer the case), earning is 200% on SQ so around 48k points. The cash + points awards are generally thought to be the best value - you get much further than pure points awards and the cash is effectively purchasing half the award cost at a low rate.
LH Miles & More has best earning rate but moderate award costs (and now has $100 award issue fee ex-NZ :evil: ). You'll earn about 72k points almost enough to get to europe in economy.
If you intend doing other trips then *A status also comes into play. If future travel patterns are similar (long haul premium cabins) then both BD and LH are good options. If lots of flights on United then UA is also good (their status bonus applies mainly on UA flights). On this one trip, not enough to get *Gold on BD, but can get *Silver on both BD & LH. *Gold is what you need to aim for to get decent benefits. LH is harder to get (100k vs 54k on BD) but lasts a minimum of 2 years and status bonuses count, so easier to maintain.