Congratulations on choosing Melbourne. December is a good time to come. A bit of hot weather but generally in the 20Cs and very pleasant.
20 days isn't much for all three places. I haven't been to Fiji (but you might want to consider that for a few days on your way back so you can return home with a nice tan) but have been to NZ.
In NZ I loved Mt Ruapehu and the east coast drive from Napier to Wellington on the north island. On the south we didn't go south of Christchurch but I thought the Marlborough Sounds area was lovely. There are great art galleries/museums in Auckland, Wellington and Christchurch. Of course, the south of the south island is famous and you should see Milford Sound, etc. if you have time. Some people just stick to one island, on the basis that they'll go back another time and see the other. There are direct flights to both Melbourne and Sydney from Auckland, Wellington and Christchurch.
As far as Melbourne is concerned, you should see (in my opinion and depending on your interests):
- Botanical Gardens have a stunning collection of flora (and you can go to the cinema at night). Also have a coffee at the Observatory Cafe (NOT the cafe in the middle of the gardens) which is delightful and modern and next to a good gift shop. It's also good for people watching as it's a centre for runners who jog "the tan" around the gardens
- Melbourne's other gardens - the Fitzroy (also with nice cafe), Treasury and Exhibition Gardens are all lovely for a picnic
- the Shrine - Melb's enormous 1920s-built memorial to casualties of war with marvellous recent extension and great views from its balcony - it's over the road from the Observatory Cafe
- Federation Square is the modern Guggenheim-esque development which houses the Australian art collection of the National Gallery of Victoria - a must for the architecture (love or hate it) and the art
- National Gallery of Victoria - international collection - just down the road (the beautiful boulevard St Kilda Rd) from Fed Sq. This collection punches above its weight in international art because of the "Felton Bequest" which allowed it to get into the market in the early 20C and buy spectacular works that other public galleries couldn't afford at the time. Recently renovated and well worth a visit.
- Government bldgs - Parliament House, the old Treasury are all at the eastern edge of town and grand Victorian bldgs, the latter now with a museum which is meant to be good
- Melbourne Museum and Royal Exhibition Building - the museum is OK but the precinct is interesting with grand Victorian and modern architecture juxtaposed. The Exhibition Bldg is where Australia's first federal parliament sat (Canberra wasn't built until the 1920s).
- The Zoo - Melb has a good one and it's just north of the city
- Lygon St is commercial but popular with tourists. It's got lots of Italian restaurants if you like Italian food. If you want somewhere for younger and groovier people - try St Kilda, Brunsick St in Fitzroy or Smith St in Collingwood.
- St Kilda is where groovy young people go. Lots of cafes and bars and a gentle bay beach which is pleasant for swimming. Lots of backpacker accommodation too. I think it's a bit grungy - but I'm probably middle-aged and boring
- Like everywhere else, Melb's Docklands are being redeveloped. I haven't been there myself but if you're interested in urban renewal, it might be worth a look. You can do river cruises etc (I have never done one) but that might be a good way to see the riverside and dock vista. I imagine the cruises with food are exxy, so try to do one with just sightseeing. Alternatively just walk along the Yarra River and see it all on foot.
- The cathedrals - St Pauls in the city centre and St Patricks on the eastern edge. The former is sandstone and the latter the bluestone so characteristic of Melbourne. St Pats is wonderful gothic revival
Plus ... go on a tram - they have lovely old ones circling the city centre that are free; have a coffee - Melbourne is cafe central; buy some clothes - Melb is a great centre for young designers and their boutiques.
Re accommodation, I don't know because I live here but try this website which is designed for people travelling with your budget. travel.roughguides.com/destinationshome.html has a wide variety of options, depending on where in Melb you want to stay. Just follow the dropdown menus. Of course, this is the home of Lonely Planet travel publications and you should consult them too. Hope this helps.