I guess it depends on what sort of scenery you would rather see. The landscape in Tasmania can vary greatly from region to region. My favourite driving route is along the Tasman Highway heading to the north-east. You start in Launceston and drive in the direction of Scottsdale, and from there you go to Derby, after which point the countryside tends to become brilliantly green and gorgeous and more than slightly hilly. You follow the same road to Pyengana, which is located in an amazing sweeping green valley, and from where you can access waterfalls and bushland and so forth. If you keep following this road you eventually get to St Helens, which is a much more developed little town on the edge of the coast. I would recommend this drive if you are interested in seeing rich hilly farmland, literally ancient myrtle trees, bushland and valleys! :-) It also tends to feel less crowded than some other areas.
You could also hang around the Launceston region and visit Liffey Falls - there is a national park here and a lot of hilly forested areas - Mole Creek - pretty farmland and the Trowunna wildlfe rehab centre - Deloraine, Westbury, Evandale and related little towns. This is very much an agricultural area as well but there is also a significant amount of forest and bushland.
If you would rather be nearer the water then the Burnie and Devonport region provides yet more hilly farmland, but with "ocean views"...
More thoroughly "Australian" bush settings can be seen going north towards Georgetown and Beaconsfield and in the north-west going towards Smithton.
Another route is the drive south from Launceston to Hobart through flatter, often yellower (depending on the time of year I suppose) country. South of Hobart the hills become quite dramatic again on the way down into Huonville.
If you want to see the south-west wilderness then it is perhaps best to be based in the Hobart area. You cannot access the wilderness by car, but I think the closest you can drive to it is a place called Strathgordon, which should be marked on any map of Tasmania. I would imagine that it is definitely worth visiting if you are interested in seeing unspoilt and truly wild wilderness.
Another region is the central "highlands", where the country is different all over again, strangely isolated and with a distinct atmosphere. This area is found partly in the Walls of Jerusalem National Park, which should be marked on a good map.
In any case, I think it is definitely necessary to have a map, though perhaps not always to stick to it; there are innumerous little out of the way places that are often even lovelier than those boasted of in pamphlets and related advertisements. I hope you have a fun time. I am sorry that I could not be more specific or provide a slightly more coherent response!