re NZ and Oz - some favourites you may like to try are:
NZ -
auckland domain - wintergarden - glasshouse/fernery complex - edwardiana
auckland regional botanic garden, south of the city
wellington - botanic garden, in thorndon, in the city
wellington - Otari native plant garden (need a day here!)
new plymouth - pukeiti rhododendron trust /native bush as well as rhodos galore, a day trip
new plymouth - Tupare garden, large private valley garden
new plymouth - Pukekura Park, in the city
Christchurch - botanic garden

Oz - Canberra - Australian National Botanic Garden - natives only - need a day here too
Canberra - gardens of Old and New Parliament House (insist on a courtyard tour of the New - v. interesting)
Canberra - sculpture garden of the National Art Gallery (all native and quite something)
Canberra - Commonwealth Park, north side of lake
Sydney - Botanic gardens - need all day here
Sydney - Bicentennial Park - brand new parklands next to the site of the 2000 Olympics at Homebush Bay
Syd - Mount Tomah botanic gardens - cool climate plants including interesting Gondwanaland garden and formal use of Oz native plants
Syd - Mount Annan bot gardens - natives only
Mount Wilson, Northern Blue Mountains - many private gardens, very lush hill station - atypical for Australia
Cairns - botanic garden in the tropics
Brisbane - botanic garden in the subtropics
Brisbane - Mount Coot-tha Botanic Gardens - relatively young and fascinating
Cooktown - botanic garden in the hot tropics!
This list looks very institutional - apologies, there are thousands of private gardens in both countries which are open to the public, some all the time, some only for certain days etc. I have a guidebook to Oz open gardens if you want me to look up certain places - happy to - it depends what sort of thing you're interested in really - as well as horticultural show-off pieces there are many historic gardens open too, often only known about by locals...
enjoy!
Adelaide - botanic gardens
Perth - King's Park, botanic gardens (not to mention wildflowers)
Hobart - botanic gardens
Melbourne - Royal Botanic Gardens
Melb - Fitroy Gardens
Melb - Cranbrook botanic gardens - new and native only
Mount Macedon - half an hour's drive north west of Melb - a Victorian hill station with lush, bush girded gardens, &
the Dandenongs, similar but to the south east...
Dromana garden, large private garden on the Mornington peninsula - about 45mins drive from Melb -
Rippon Lea, suburban Mansion garden - boom era gold/merchant's paradise Victoriana - National Trust icon

Thailand/ Bangkok - visit Jim Thompson's house - not far from the city centre and quite charming - the house is worth a look too - tropical plants and intimate scale, water much in use /blended traditional thai architecture/ silks and antiques galore - a heady mix and an oasis in a busy city...

Vietnam/Hanoi - visit the Temple of Literature - a 1200s university with again very tranquil courtyard gardens, in the middle of the...

eVEN IF YOU MAY BE BACK ALREADY I will give some informations about Asian gardens:
Nepal: Godhavari Botanical Gardens near Kathmandu are worth visiting because you will get a good introduction into the tree flora of Nepal-Himalaya midmountain areas. Trees are labelled, and there is (sometimes) a garden plan avaiable.

India: Darjeeling Lloyds Botanical Garden was a very good one just until say 10 years ago. Unfortunately now more and more houses are built into the gerden area. This is a wonderful location for the eastern himlayan flora, but not very well meintained any more and the plants are only sporadic labelled. There is a good garden plan to buy if you are lucky enough to find somebody who can sell this.
India: Dehra Dun: Forest Research Garden nad Arboretum is very interesting to those who are keen in dendrology.
India: Ooty Botanic Gardens is a well known place with lots of mid-indian plants, well preserved and maintained, good labelled.
India: Calcutta Botanical Garden: You have to see it.
some more gardens in India, you will get informations in Dehra Dun or Calcutta.
Indonesia: Bogor is a must. I have been there 4 weeks ago (end of July/beginning of August) and all Java is very save for tourists - do not bother about riots-tzhey took place in Jakarta and Solo. People love to see foreigners now - nobody is travelling there since May.
Cibodas (alt.: 1200-1400 m ) east of Bogor is a good place for mountain species, mainly on trees, lots of Gymnosperms. It is an branch of Bogor.
Bali: near Bedugul in Central Bali 1200-1400 m altitude, trees and plants from eastern Indonesia, Bali, Lombok, Nusa Tengara Timor etc., 140 ha , but not as good labelled as Bogor and Cibodas.

Once travelling through these areas all are worth a visit, but outstanding are Calcutta and Bogor.