Oct/Nov is a bit early for NZ, unless you like cool weather. Temperatures for this week in Auckland have highs in the low 60s. In Dunedin you are looking at the low 50s. You might get better weather in late November. Jan, Feb and Mar are the best months for NZ, but nothing is guaranteed. Dec/Jan is summer school holidays in both countries.
For Oz anytime between Oct and Apr is good. Dec-Mar are the hottest months.