Although I live in Canada (Montreal) and come from the states (Pennsylvania), I love Australia and Melbourne in particular.
There is the Victoria Market - a must visit. and the evening Suzuki market in the same location with local restaurateurs and a very interesting flea market. ( I bought two antique hand sewn Kimonos there for around ten dollars US each... )
I've stayed right on the boardwalk (expensive) and further south in a boutique hotel on the tram line to the beach.
There is a free tram zone for tourists, which stops at most of the great places to visit, like the train station and the Victoria market, and the art museum, so anywhere with access to the tram line is the place to stay.
Renting a car would open great possibilities but we are Canadian, American, Greeks who drive on the"other" side so relied on pubic transportation and access to city center to get around.
I've stayed at the Royce Boutique Hotel which is on a great tram line, and at the Crown Casino, which is more expensive than its worth, although a walk up and down the boardwalk at sunset is definitely worth doing.
I think I'd recommend the Royce Boutique Hotel or one along the same tram line.