The laneways — Hosier Lane's ever-changing street art, Degraves Street's cafes, and arcades from the 1800s. Free, and best explored on foot with no plan. This is the Melbourne experience, more than any ticketed sight.
Great Ocean Road day trip — the coastal drive to the Twelve Apostles sea stacks is one of the world's great road trips. Full-day tours run about A$115-170 ($80-$120). It is a 12+ hour day; the payoff at the Apostles is worth it.
Phillip Island penguin parade — every sunset, hundreds of little penguins waddle up the beach to their burrows. Entry about A$30 ($21), or A$120-160 ($85-$110) with round-trip tours from the city. Especially memorable with kids; dress warm even in summer.
Melbourne Skydeck — the southern hemisphere's highest viewing deck, 285 metres up the Eureka Tower. About A$31 ($22); the 'Edge' glass cube that slides out of the building costs extra. Go on a clear day — check the sky first, not the calendar.
Queen Victoria Market — a working, 145-year-old market for cheap eats, fruit and souvenirs. Free entry; closed some weekdays, so check days before you plan around it. The night market (summer and winter seasons) adds food trucks and live music.
Melbourne Zoo or SEA LIFE Aquarium — the zoo (about A$42 / $29) has koalas and kangaroos 4 km from the centre; the aquarium is the rainy-day pick with king penguins. Choose one, not both, on a short trip.
NGV International — Australia's most-visited art gallery, on St Kilda Road. General entry is free; only the big touring exhibitions are ticketed. The stained-glass ceiling in the Great Hall is worth the walk alone, and it's the top rainy-day pick.
St Kilda — the bayside suburb, about 25 minutes by tram: a pier, cake shops on Acland Street, and a small colony of little penguins at the breakwater at sunset, free to watch. It's a fraction of Phillip Island's parade, but it costs nothing.
Puffing Billy — a century-old steam train through the Dandenong Ranges forest, about an hour east of the city. Returns from Belgrave cost around A$60 (~$42). It's the day trip for families with young kids — shorter and cheaper than the Great Ocean Road.
Skip a paid city bus tour — the free City Circle tram (Route 35) loops the same landmarks with commentary.