Question regarding using a new passport for the first time?

hi guys. my old passport expired mid last year. i got a new one from sydney passport office. but i never used the new passport. im planning to go back home this easter break. just wondering if i need to carry the old passport along with me to show a connection between two passports?

Does new zealand immigration automaticly shares information with au immigration? eg this person is now holding a new passport.

What do you guys do after u get ur new passport? Is it as simple as buying a ticket and fly home? Please let me know, thx a lot :)