Public Transport Victoria — for train and tram timetables, ticket info, etc.
That’s Melbourne — a ‘what’s on’ events guide.
Visit Melbourne — the city’s official tourism website. I won’t be too offended if you use it (just don’t expect me to be nice to you if we ever meet in the street).
Google map of Melbourne — because everybody needs a map.
Bicycle Network — maps and lots of handy info on riding bikes in Melbourne.
Bikemap.net — more off-road bike path maps.
Jobs & classifieds:
Seek.com — for jobs.
Gumtree — for jobs, housing, buying/selling, and other bits and bobs.
City of Melbourne — city council website containing lots of nuts and bolts info.
Bureau of Meteorology — for Melbourne weather forecasts.
Wikipedia’s Melbourne page — for a whole lot of info.
Handy sites for travellers:
Oanda — currency converter.
Google Translate — useful for obvious reasons.
HostelBookers — again, fairly obvious.
Hostelworld — obvious, again.
CouchSurfing — because it’s brilliant.