For a largeish city Melbourne is remarkably safe. Leaving aside the drunken brawling that's become a regular feature outside  many of its nightclubs, there's really little in the way of any serious risk to the average visitor. Street crimes like assaults, robbery and even petty crime like pickpocketing are very rare and even more infrequently directed at tourists. In fact Melbourne is so safe it verges on the dull ( not that safe or dull are necessarily bad things).

Be aware that some Melburnians have a rather black sense of humour and will often invent imaginary perils to worry tourists and amuse themselves. Killer UV, enormous snakes, drop bears etc. Take it all with a large grain of salt but of course don't overdo the salt that's bad for your health. 

The weather can be somewhat changeable, especially in winter and early spring, so bring some warm woolies and a brolly.