Without a doubt, the best way to see Austria is to hire a car, and drive yourself through the country. The roads are all excellent, and often you find yourself stopping frequently, particularly in the Alps as you photograph the amazing vistas.
Otherwise, it is relatively simple to get around Austria by using the trains. These are clean and good value with connections available throughout Europe. It helps to speak some German when ordering tickets, although most counter staff speak English.
In Vienna, there is a good tram network, and daily travel cards are quite cheap. It is also possible to buy a Vienna Pass that allows access to some museums as well.