There is a lot to explore in Melbourne, within the city as well as outside of it. Public transit is fairly easy to use and accessible once you get the hang of it. Have heard it to be the culture capital of Oz, and it is indeed. Many artistic establishments open to the public. Lots of cultural diversity. Outside of the city itself are the Great Ocean Road (on a nice day, just magnificent views), the Dandenong Ranges, and Prince Phillip Island (known for its nightly parade of penguins).
Insider Tip:-Save yourself a lot of hassles by using a myki transit pass. Some hostels will lend you a myki card free, but you have to put money on it. Getting around is so simple with myki. -Save money by taking the free walking tours of Melbourne. There is a free walking tour you can book at Flinders Stn. visitor ctr. or online, but book ASAP, they fill up quick. Also good tour with www.imfree.com.au. Technically not really free, since guide will ask for tips at the end, but still good value.