Four days in Hobart: a mountain above the town, and a gambler's museum under the ground

Start on Salamanca Place, a single row of sandstone warehouses facing the docks. They were built in the 1830s to service the whaling fleet, which is what Hobart ran on before the whales gave out, and they now hold galleries, bars and workshops behind the same facades. On Saturday mornings the entire street becomes a market, three hundred stalls of Tasmanian timber, produce and craft, and it is the best morning of the week to be in the city.

Behind the warehouses, a flight of stone steps called Kelly's Steps cuts up the cliff to Battery Point. The suburb takes its name from a gun battery placed on the headland in 1818, and it is one of the oldest surviving subdivisions in Australia: narrow lanes, weatherboard and brick cottages, and a ring of tiny Georgian houses around a green called Arthur Circus. Almost none of it should still exist. It survived because residents and unions blocked the demolitions in the 1970s.

Wander down through it to the waterfront for dinner. The fish punts moored at Constitution Dock are the local answer and cost about a third of what the restaurants behind them do.

The gardens go first because they are on the way to the pier and they open early. Established in 1818, they hold six thousand five hundred species including four hundred Tasmanian ones, and the thing to find is the Subantarctic Plant House, where a chilled, dripping, wind-blown glasshouse recreates Macquarie Island using plants collected fifteen hundred kilometres south of here. Nowhere else has one.

Then take the ferry. MONA sits on a sandstone peninsula about twelve kilometres upriver, and the boat from Brooke Street Pier is a proper part of the visit rather than transport: half an hour up the Derwent past the zinc works and the wooded banks, on a catamaran fitted out with painted sheep to sit on.

The museum was built by David Walsh, a Hobart man who made a fortune from mathematical gambling systems, and it opened in 2011. Almost all of it is underground, cut three floors into the rock, entered from the top so that you descend as you go, with no labels on the walls and a phone app instead. The collection runs from Egyptian mummies to work designed specifically to offend, and a fair number of visitors leave annoyed, which the founder has said is the point. Give it five hours and eat there.

There is a winery and a brewery on the same site, both good, and a lawn above the river to sit on afterwards.

One road runs south west out of the city up the Hobart Rivulet, and everything today is on it.

The first stop is the hardest. The Cascades Female Factory operated between 1828 and 1856 as a place to hold, punish and reform transported women, and thousands passed through it. What survives is mostly walls: yards, a nursery where infant mortality ran appallingly high, and the outlines of the buildings. It is one of eleven convict sites on the UNESCO World Heritage list, and the guided tours are what make it comprehensible, so time your arrival to one.

A few hundred metres further up is the brewery, which has been making beer on the same spot since 1824, longer than anywhere else in Australia, in a Georgian building that looks more like a country house than a factory. Tours run about ninety minutes; the heritage version takes children and the brewery floor version does not.

Then keep going up. The summit road climbs the whole 1,270 metres in about twenty minutes of driving, through wet forest and then out onto bare dolerite boulder fields, and there is an indoor lookout at the summit facing down over the whole city, the estuary, the Tasman peninsula and the mountains inland. Take the coat you would take to a Scottish hill in November.

Hobart has been the departure point for Antarctic expeditions since the 1830s, and it still is: the icebreakers tie up at the waterfront and the Australian Antarctic Division runs from just outside town. The state museum, five minutes from the dock, has a permanent hall on it, along with Tasmanian natural history, colonial art, and an unflinching account of what happened to the island's Aboriginal people.

It was established in 1843, it is free, and it occupies a run of the oldest buildings in Hobart, including a commissariat store from 1808. Two hours is right for a last morning.

Then walk the last stretch of the waterfront before the airport bus, and look at the mountain once more on the way.