In Australia the easiest mountain to go to without a car is Mt Buller, as there are buses directly from Melbourne Airport (Takes 3-4 hours, and is the closest mountain to a major airport) up to Mt Buller and on the mountain every thing is located within walking distance or there is a free shuttle bus around the mountain. There is a YHA, youth hostel up there which I believe has cheaper accommodation but i'm not sure.
Or else, I think the NSW resorts have a small airport near them and you can get a helicopter to falls but I don't know much about that.