I would say you could actually have a car in Sydney, it may end up cheaper to take a two week rental than just having a rental for a couple of days here and there. You're looking at around AUD1,000 for a two week rental plus fuel and tolls.

There are plenty of city motels that you could stay at which allow parking for free and are still pretty central. The Twin Towers in Artarmon is one.

As for route you could easily go SYD - Blue Mountains - Goulburn/Canberra - Geelong - GOR - Melbourne - Phillip Island - Bairnsdale - Mollymook - SYD in about 8 - 10 days. The one word of warning is that these roads are more PCH than Interstate in quality.