Hi Whirly,
Getting to YSBK is do-able by public transport. There's a number of trains to Bankstown (er....the town of, that is, not the airport of...). There's a bus stop on one side of the train station (forget which side) with buses that will take you right past the main gates of the airport on Marion St.
Some schools aren't on that side of the field however so you can either walk around the perimeter road or try for a bus route that passes the opposite corner of the field. That would be along Milperra Rd and you'd need to alight at or near the intersection of Milperra Rd & Henry Lawson Drive.
Alternatively there's heaps of taxies at the train station. Takes about 5 or 10 mins & should cost around A$10-$15.
Definitely try for a whiz over the harbour bridge/opera house. Victor 1 @ 500' past Sydney International can be fun too. If you like more rugged terrain then west to the Blue Mountains is nice.