Hobart is the southern end of Australia, and it knows it. It was founded in 1804 as a British penal colony, which makes it the country's second-oldest capital after Sydney, and the sandstone warehouses on the waterfront still look like the whaling port they once served. Behind them rises kunanyi / Mount Wellington, 1,271 meters of dolerite cliffs that can hold snow even in summer. The land is nipaluna to the Muwinina people, who lived here long before the first ships arrived, and Aboriginal Tasmanians may have been here for as long as 35,000 years.
The city itself is small and walkable, and you can see most of it in a day and a half. The reason to stay five days is what surrounds it. Within two and a half hours you can reach a UNESCO convict prison, a bridge built by convicts in 1825, an island that lives off oysters and cheese, and one of the most photographed beaches in the country. Add Mona, the private museum that turned the city's reputation upside down when it opened in 2011, and Hobart becomes a better base than most capitals.
The plan keeps Hobart to the first two days and gives each of the next three days to one trip. Day one is Salamanca Market, Battery Point and the waterfront, day two is Mona by ferry and sunset on kunanyi, day three is Richmond and Port Arthur, day four is Bruny Island and day five is Freycinet and Wineglass Bay. Start on a Saturday: Salamanca Market only runs on Saturdays, and Mona is closed on Tuesdays and Wednesdays, which this order avoids. Read the Australia country page for the bigger picture, and see the guides to Sydney, Melbourne and Adelaide if you are building a longer trip. You need a car for days three to five, or a day tour for each.