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Melbourne’s CBD (Central Business District) (ie: Melbourne proper, a rough equivalent of a US ‘downtown’, kinda like a smaller Mitte for Berlin-familiar folk) is the centre of everything, and it’s easy to get almost everywhere. It’s where I live, and I love it. Plenty of food and drink options here, and especially in the east half, not too crazy at night (King St, on the west side, is home to stripper joints and the like, not my sort of thing, hence I mostly stick to east of Elizabeth St).
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Fitzroy, Carlton, Collingwood are the trendy/hipster suburbs, with plenty of restaurants, bars, cafes. Perhaps something along the lines of Chicago's Wicker Park / SF's Mission / Berlin's Kreuzberg (though less gritty). Fitzroy in particular is pretty easy to walk around, and is worth checking out even if you’re staying elsewhere.
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St Kilda: very trendy these days, and right by the beach - albeit on Port Phillip Bay, so don’t expect much in the way of waves. If you want a proper Australian beach experience, it’s best to hunt that down in Sydney (Bondi, Manly, Bronte, etc), as the better beaches near Melbourne are probably a good hour or two drive out of town.
North Melbourne, South Yarra and Richmond are nice enough too, and it’s not too much of a hassle to get from any of those into the CBD via public transport (a Myki card - our Ventra/Clipper/Oyster/Octopus equivalent - will work across all trains, trams and buses), and Brunswick is cool too, albeit a bit further out, but I’d recommend either CBD or Fitzroy if it fits your budget.
I cover a decent amount of options on Foursquare: https://foursquare.com/user/124286/list/melbourne-australia
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Melbourne Supper Club - probably my most favourite place in Melbourne, a classy bar with excellent snacks & desserts and comfy couches, plus they’re open until the very early hours of the morning. Bonus points: inside, thus no smoking, and the music isn’t stupidly loud. The creme brûlée is the best I’ve come across.
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Spring St Grocer - owned by the same crew as the Supper Club, just a couple of doors away, and it’s the best gelato in Melbourne, rivalling the best I’ve found around the world (including Italy).
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The Everleigh - generally considered the best cocktail joint in Melbourne (and there are plenty of contenders). Great booths, friendly staff. Bartender’s choice is almost always the way to go (especially for non-alcohol drinkers like myself).
Great Indian at Red Pepper, decent fancy-Mexican at Mamasita (well, it’s quite good by Australian standards, but Mexican isn’t nearly as common here) - treat it as a late night option though (9:30pm or later), always a line at normal dining hours. I’m a fan of the Japanese at Gyoza Douraku, even though the J-pop is sometimes a bit too loud, and the Chinese dumplings at Hu-tong are excellent. The options are better for vegetarians at Shanghai Street though (including veggie xiao-long-bao). The menu at Fitzroy’s Trippy Taco is entirely vegetarian.
If you’re a coffee fan, there are many great options - it’s joked that Melbourne’s biggest export is its baristas, and there is some truth to that. In the CBD, Brother Bubba Budan, Patricia, and The League of Honest Coffee all have good reputations. In Fitzroy, try Proud Mary, Everyday Coffee or Industry Beans. Excellent brunch places are also pretty easy to find too - especially in Fitzroy - some of my best friends live there and can provide up-to-date suggestions if needed.
February 20th is White Night, which is in its fourth year in Melbourne (borrowed from France’s Nuit Blanche), with music, light displays, odd installations and more taking over the CBD and its fringes, pretty much from 7pm to 7am. The CBD will be absolutely packed that night, which makes getting around a little challenging. Still, it can be fun to wander around and take some of it in, but I’ve found it’s a bit easier to deal with after midnight.
If you need a desk and/or wifi, the coworking space Inspire9 is a great place to stop and work for a day. It’s rare that Australian cafes have wifi, so an unlocked phone with a local SIM and tethering is highly recommended. Telstra is our formerly-government-run carrier, and while their prices are higher and I’m not a fan of their anti-competitive behaviour, their cell network is still clearly the best.