Despite Australian cuisine lacking much of an identity in itself, a travel destination like Melbourne tends to make a name as the gastronomic equivalent of the United Nations, accommodating delicious fare from across the globe (without the political ramifications). Anywhere you go in Melbourne, you'll tend to find the British staple of fish and chips, the Japanese staple of sushi, and international stand-by's like pizza, diner/pub grub, and beyond.
If you're one for Asian food, Melbourne has a large stable of Chinese food operators throughout the city, although they are particularly concentrated around Little Bourke Street in the Central Business District. Vietnamese can be found on Victoria Street in the Richmond neighborhood, while you don't have to stray far from downtown to satisfy other Asian food cravings.
Your best chance for enjoying Melbourne's historic coffee and cafe culture is along the university strip of Lygon Street in Carlton, which also provides a home to a ton of high-level Italian eateries. There are also plenty of restaurants of all varieties in the tourist stronghold of St. Kilda and in the nearby community of Geelong, while you could legitimately be satisfied with never leaving the CBD due to its variety and quality of eating out options.
Long story somewhat summarized? No matter your preference for grub, you'll find it in Melbourne. You won't have to eat kangaroo burgers around every turn; although once you try one, you may want to.