if u r flying united, you must go thru sydney. Qantas has direct flights to melb , syd and brisbane.

ALso, both have 3 flight airpasses that r best value. (allows 3 internal flights in oz). FLying b/w spots is best due to 14 day time limit (not enough in my mind but the US sucks like that. If its ur first and possibly only time, haggle ur boss for more time. u'll appreicate it.

Im from melb living in LA and have done flight about 40 times.Melb suggestions would be Great Ocean Rd (4 hr drive), Healesvile Sanctuary (best open / walking zoo).
Go around the city a bit too. Skip the casino - its crap.
To get around city public transport is best - can be expensive but daily tix r best. Melb has great pubs and food everywhere.
Lygon st is awesome vibe for food etc.

Dec is getting into summer. At least walk around Melbourne Cricket Ground (aust sporting mecca) - but i dont suggest going to a cricket match unless you are REALLY keen, and Flinders park ( Aust Open Tennis ) site. Also, Formula 1 GP held at Albert PArk ( 10 mins south of city).

Hope this helps....
Try a pie, all the beer ( BOAGS), fish and chips is a MUST.
Burgers with the lot, a chicken parma at a pub.
(Chicken parma @ palmerston hotel is legendary).

if Captain Spalding band is playing go see them..they are great!