I am a new graduate registered nurse. Is it necessary for me to apply for a malpractice insurance?

Is it all right to join unions and state nurses' association?

Congrats on passing your boards! Hurray for you! YES it is absolutely 100% necessary to get malpractice insurance!!
If something were to ever happen (god forbid) and you went to court not only is the patient that is suing you (and their lawyer) looking for an excuse to sink you, but the hospital does not want to pull out the big $$'s on your behalf so they are looking for actions in your charting which were not according to hospital policy.
For example, you have been caring for a patient with CHF, who is on a fluid restriction. Hospital policy states that when you flush an IV site you use 3 mLs, you used 10 mLs every time you flushed, which say was 10 times that day. Well, patient died and you're in court because the family says he died because he had too much fluid . . .you did not follow hospital policy in only flushing with 3 mLs thus the hospital will throw you to the wolves and you're SOL.
Bottom line is you worked REALLY hard and you must protect yourself and your license at all cost. The hospital is not looking out for your practice, they are looking out for themselves. For the price of carrying your own insurance, it is worth it . . . especially if you are on the stand one day . . . by yourself.

Congrats on passing. It is only necessary if you wish to work in that field! It is especially important if you want to work in North America where there is an excess of lawyers.