If you're staying near or in the city then you won't need a car, public transport is really good apart from the trains not always running on time!
There's lots of places to go, depends what you like really.
Acland Street, St. Kilda - shopping, luna park, good food and music, market on sundays, best place to go to the beach on a hot day
Chapel Street, Sth Yarra- shopping, clubs, Jam Factory, pubs, good music
Flinders lane, little collins street, city- shopping, alternative shops, clubs
Bourke street, city- myer, david jones, metro, cinemas ---- main city shopping mall, good music shops etc
Lygon Street- Italian district, great food, music and drink...
Albert park lake/Melbourne Sports and Aquatic Centre- basketball courts, swimming pool, albert park lake is where the aus grand prix is every year..
I'm not too sure what you're interested in but there's a bit of everything there....
reply back and tell me what you want to do and i'll tell you the best place for it
xxx