Transport from WLG airport to the CBD is pretty simple - there are a few options.

There is the 91 "airport flyer" which leaves 50 meters from the terminal building (go past baggage claim and out the door to the right) and head towards the light! Its I think $8 cash to the driver and even in peak traffic, should be in the city in under 20 minutes. If you want to super shuttle it (shared ride), its about $20 door to door. The shuttles leave when full or about 10 minutes after the first passenger turns up so not too bad. Otherwise a cab is about $40 into the city. The random option (if you are willing to walk for 5 or so minutes) is to head out of the terminal to catch a local 11 bus into the city. Takes a bit longer but gets you out of the bustle of travel straight away.

Enjoy your time in the Capital city.