Information current as of March 2023
Sydney
cheap
take 420 from International, bus stop outside terminal opposite Rydges hotel, 420/350 from Domestic (bus stop outside Qantas terminal 3) then change at Mascot for train to city
350 bus runs from Domestic (Qantas Terminal) to Mascot Station then eastern suburbs (ends at Bondi Junction) - this one is good for UNSW.
420 runs from Mascot Station to Domestic then International then Burwood (trains at Mascot, Banksia or Rockdale)
faster
train stations directly at International and Domestic, cost is around $18 to city (whereas from Mascot station cost is around $3-4: addition is airport surchage)
Opal card or credit card can be used (cannot purchase Opal on bus)
Melbourne
cheap
901 Bus (Frankston) leaves from the ground floor parking lot for T4, short walk from T3. Myki needed (there used to be machines at Melb airport, not sure. This method is more for locals)
Take 901 to Broadmeadow Station (15 minutes or so) then a metropolitan train to the city
faster
Skybus direct to southern cross station, return ticket (valid for 3 months) $32 return. Buy ticket online (smartphone) or at Skybus stops