You can do the coastal route using public train/bus and it is quite cheap fare, but takes a long time and some routes run once or twice a day. But well worth doing as it is quite interesting scenery with charming small towns (few thousand people each for the most part). I'd do Sydney to Narooma the first day and stay overnight there. But it illustrates a problem with no car, Tilba is a few km away and well worth a visit but doubt it can be done with a bus. There is a bus from Narooma that connects to a train to Melbourne (that takes about 12 hours). Same would be true of towns all along the south coast so you could stop at Bega, etc.