If Australia is unique, Tasmania is probably the uniquest. Although it is the smallest state in Australia - and an island state in itself (there's actually around 1,000 islands making up Tasmania including the Bass Strait islands and Macquarie Island in the Sub Antarctic) - Tasmania has a rich but sad colonial history and wonderful diversity of wildlife, scenery, natural wilderness areas and a relaxed pace of life which makes it the envy of many of Australia's mainland destinations.
Tassie's colonial history dates back to 1642 when the island was named "Van Diemen's Land" by Dutch seafarer Abel Tasman. It wasn't until 1803 that the British acted to establish a presence on the island with a penal colony established on the River Derwent (near where Hobart is today). It turned out to be an unfortunately sad and violent start, with many issues facing the settlers in the opening decades including resistance from local Aboriginals which led to many clashes and killings during the first few years (see the Indigenous Australia website), until the island was virtually "cleansed" of its Aboriginal population by the mid 1800's.
As a part of it's sometimes brutal history, a convict prison was established at Port Arthur in 1833 and was one of the harshest prisons of the time, but today the ruins of Port Arthur are one of Tasmania's major tourist attractions, located about an hour or so from Hobart. The site receives well over 250,000 visitors per year, and offers a range of historical tours (including "ghost" tours), as well as offering great opportunities to relax and explore the history for yourself. Certainly it is well worth the visit!
The best way to see the state is undisputedly by either combining car and accommodation or travelling via motorhome (but even if you have a motorhome, there are still some accommodation options you will definitely want to book into for a night or two!). Either way, you can fly down and hire one, or take your own on the Spirit of Tasmania ferry which leaves from Melbourne to Devonport most days. Tasmania itself measures only 296km by 313km at the widest points, so a "drive yourself" holiday is the ideal to get out and explore - and the distances are reasonable. However if you would prefer to be free of the worry of driving yourself, there are a number of operators running a range of coach touring options of the state if that is more your style.
Tasmania literally bursts with natural wonder, colonial charm and surprises. The beautiful capital city of Hobart sits alongside the Derwent River - with it's colonial heritage, constitution dock (site of the finish line for the annual Boxing Day Sydney to Hobart yacht race), the Salamanca Markets, the Cadbury factory (yum! Although they no longer do guided tours, the shop is well worth the visit, often with some great "factory second" bargains!), or simply get out and explore the sights. Mount Wellington (1270m) overlooks the city with some awesome panoramic views, is often covered with a dusting of snow in the cooler months, and is a great little drive to explore.
Out of Hobart, the state offers some wonderful scenery and experiences. Launceston (the other "capital", usually pronounced "Lon-ses-ton" by locals) sits to the North of Tasmania and is only about 2.5 hours drive from Hobart through wonderful colonial towns like Richmond and Ross, or take an alternate route through New Norfolk and up through Meina which is an interesting alternative (although a bit more "off the beaten track"). Launceston itself is a great springboard location to explore many of the areas in and around the North / East of the state.
However, probably the best itinerary suggestion to explore Tasmania is to base a couple of nights around Hobart to explore the area (2 nights at least - checking out Hobart and it's surrounds as well as taking in some of the greater Hobart region including Port Arthur), head up the East Coast, checking out Freycinet National Park (1-2 nights) and up to Bicheno (1-2 nights) & St Helens (1-2 Nights), then across to Scottsdale and onto Launceston (at least 2 nights). You can then continue across the North of the state, with towns like Deloraine, Devonport, Burnie and Stanley (famous for the Stanley "Nut") all worth a visit.
An absolute must stay is in the Cradle Mountain - Lake St Clair National Park area towards the Western side of the state (plan at least a couple of nights here), with a number of accommdation options springing up in the area in recent years. It does take a while to actually drive there, but it is an amazing wilderness area, with a number of hiking trails, bushwalks and some amazing scenery to behold. I personally have a soft spot for Cradle Mountain Lodge as the accomodation choice, but as mentioned there are a number of accommodation options to suit most tastes and budgets, with some amazing options available for the romantics.
After Cradle Mountain, head down the West coast and stay a few nights at towns like Queenstown and Strahan, which are gateways to the wilderness areas such as the Franklin - Gordon Wild Rivers National Park. Some great Franklin and Gordon River cruises and experiences through the wilderness areas which are a must for some amazing wilderness experiences. The Franklin is actually the birthplace of the Australia's original Green movement in the 1980's where the country's largest conservation battle took place against the proposal to build a dam and hydro electric scheme along the river. The greenies eventually won, and now the area is thankfully a part of the Tasmanian Wilderness World Heritage area.
Further South you can head to places like Strathgordon and down to the Southwest National Park, which has some great wilderness areas with huge forests and lots of opportunities for hiking and getting out amongst the wilderness. From there, you can get across to New Norfolk and back down into Hobart, and then across to Huonville and the Southern coastal areas.
All in all, Tasmania is a state that you just have to get out and explore. There's no other way to do it. The variety in scenery and experiences - from sophisticated wining and dining, through to absolute wilderness - will leave you enjoying every minute and every kilometre of this wonderful state.