The first thing to say is Melbourne is very much open for business! If you're heading over, I'd suggest going on a guided walk with Koorie Heritage Trust. The tour lifts a veil on Melbourne and you start to see it through Indigenous eyes.
Although I work in the city and can bang on about the restaurants, sports, laneways and shopping, these days my Melbourne is about the surrounding neighbourhoods. I'd start in Yarraville for a movie at The Sun, where you can catch a local indie or international blockbuster. Quentin Tarantino has been known to make surprise appearances there. Afterwards I'd see what vegan frozen yoghurt bar Goji has to offer this week. Footscray was recently named the fifth coolest suburb in the world! I definitely rate the Vietnamese and African food and the rapid gentrification means there are some pretty hip cafes and bars in the area. Fitzroy has some of the best music venues and beer gardens in town, head to The Napier or Union Hotel for some world-class pub food, and hunt around at the boutique clothing stores for a bargain.
Melbourne is blessed with a lot of wonderful green space. The Botanical Gardens are right across the road from the NGV and The Buxton Contemporary, which are both excellent. Sydney may have the beaches but we have the rivers! Both the Maribyrnong and Merri Creek are fantastic on foot, bike or with a dog. If you want to see some great Aussie fauna, Melbourne Zoo is excellent. I can't get enough of the wombats.