Personally I'd pick one hotel in sydney and stay there... Use that as a base and explore forth. Switching hotels daily is a pita.

Agree with another comment about heading to katoomba on day one after a 20 hour flight in coach...(we spent the first day at the rocks, circular quay, botanical gardens-with a few coffee breaks- then dinner over in manly and back to SYD for bed.)

As others have pointed out Tas will be cold and possibly rainy. If you look up historical weather and like that experience, great. Just be aware.