If your Mom is here she may sponsor you if she can afford. You can apply direct to embassy but you need to show your evidence/requirements i.e. P500 thou more in your account, your sponsor account,letter, your accom, etc. That's what my friend did.
CAP means Competency Assesment Program in New Zealand.It is 3 months program to take by qualified nurses there and it is required by NZ Nursing council.After taking that, they will assess your wages, assigned hospital etc. You can choose any NZ Nursing council accredited school. If you are RN, with at least 500 hrs work experience in hospital, and with 7.0 grade in IELTS, and show money in your account, then you are qualified and can apply direct to NZ embassy in Manila.
If you are not yet RN you can still apply, visit nz embassy website in Manila, but you can work only as Caregiver. Like UK, the wages is different from working in hospital and Carehomes. I have friend there already working in carehome and niece in hospital. they said, it's also a nice place to live
There are some agencies but applying direct to embassy is much better.
Goodluck!