That narrows it down to only a few hundred must-try places...
Every Melburnian should try Chin-Chin in the CBD. They don't take bookings, so if you get there after 6:30pm, expect at least a one hour wait. The 'feed me' option is the way to go.
Either before (if you have a 1+ hr wait), or after, head to Eau de Vie for cocktails. It's hidden in an alleyway, no signage, no windows, look it up before you go. Try the Elysee Montmarte, it's so good.
Afterwards, either Messina for ice cream, or Brunetti just as an experience.