Yes, welcome to TT Nuria and I echo Jingili's comments some for you are going to be arriving in Melbourne at or close to peak of winter ( seasons reversed from NH for SH just in case you had forgotten), it not meaning that a few days in Melbourne cannot be enjoyed and though Australia's winters are mild compared to northern and inland Europe, if you come from where it is a bit warmer, do make sure you at least have a good warm wind and rain resistant winter coat.
It really depends on what your interests are and what you would like to see which will determine what you may want for an itinerary for cities are cities much anywhere you go even if Australian cities are much younger than those in Europe and do not have as significant older quarters but they do have their attractions both around the city and further out for which you can use public transport in some cases or get a tour for half a day to multi days.
Melbourne for instance has one of the best botanical gardens you can find and many more open gardens and parkland spaces about the central area and being relatively flat it is easy to walk about though there is also a Free City Circle Tram that you can hop on/off at regular tram stops and sometimes there is an understandable commentary as you go along, it taking about 30 -45 minutes to do a full loop and then if you want to go and look at a particular area, just keep going until you get to the appropriate stop. There are also many different areas for cafes, bars and restaurants, many more expensive ones along what is called Southbank where the Casino is but an enjoyable walk along there too, there being gas flame jets that do an eruption cycle ( on the hour I think ) and on a cool night you can feel a bit warmer just for a few seconds if you stand close enough - it also a good location on the south side of the Yarra River to look back across the river to the City lights at night.
Melbourne is arguably known as the sporting capital of Australia and one feature is a game of football which is unique to Australia, Aussie Rules as we call it or officially the Australian Football league - www.afl.com.au for some insight, a very spectacular game played on an oval shaped field with an oval shaped ball, there two main stadiums very close to the city centre - the MCG ( Melbourne Cricket Ground ) which was home to the Olympic Games in 1956 and has been redeveloped over the past 60 years to now seat close to 100,000 people, it being what you could call a modern colloseum gladatorial like stadium and then there is a newer Docklands stadium which has a closing roof for when there is bad weather, there being day and night matches at both grounds from a Friday night to over the weekend so if you happen to arrive on a day so you can take in a match, you will see something of Aussie culture re our love of sport and if at a game it will be a good place to try a Meat Pie to be washed down with a glass of beer.
For getting further away from the city, have a look at http://parkweb.vic.gov.au/ to give you an idea of the surrounds, there being things like the Great Ocean Road or Fairy Penguins and Koalas at Phillip Island for day/half day trips and much more.
Sydney as with most Australian cities is similar if with different attractions for it is built around a magnificent harbour, said to be the best on the planet and though there are places like San Francisco, Hongkong or a few other coastal locations, Sydney does not have earthquakes or typhoons and has a climate that is hard to beat. No free city circle tram in Sydney and a much more congested inner city area but they do have The Rocks which is where first settlement occurred when what is called the first fleet arrived in 1788 with convicts and marine guards so much history about there and also some lovely older hotels. In place of a city circle tram, do make sure of doing some trips on the harbour ferries, especially from Circular Quay ( the city terminal between The Rocks and the Opera House ) to Manly and then you can walk the Corso to the Pacific Ocean side and then with a Ferry to Watsons Bay, you can take a walk up to what they call South Head for some spectacular views down the harbour to the city centre.
Sixteen Days is not a huge length of time but you couls still see a fair bit of Australia if you want to, for instance:
. If 2-3 days is enough for you in Melbourne.
. Fly to Uluru - 2 days
. Fly to Darwin for Kakadu - ( 3 day camping tour ) - 5 days
. Fly to Cairns - 3 days ( GBR and rainforest tour )
. Fly to Sydney - 3 days
You can look at a number of variations and it is really just what is important to you, and what your budget is for transport and accommodation but for best flight prices do decide quickly as our internal flights usually have limited numbers of cheaper tickets and they are sold quickly. Use webjet or the other site to find cheapest tickets and then book direct with the airline.