Are there any alternate ways to get to the Australian Embassy in Bkk or is a taxi the best way to go??

It is located on Sathon South rd. A taxi at non peak will cost around 250-270 baht max includ the 50 baht surcharge

The only real alternative is the A1 bus from the airport to BTS Mo Chit, and then take either the BTS to Sla Daeng and then a motor bike taxi down soi convent OR take the MRT from Chatuchak stn to Lumpini station and then motorbike donw south sathon rd to the embassy.

http://www.bts.co.th/customer/en/02-route-current_new.aspx

http://www.bts.co.th/customer/en/images/area-map/S2_update.gif (you can see the embassy at the top of this S2 map). The MRT Lumpini station is just off map to the left of the Sukhothai Hotel

A taxi is really the way to travel.

Taxi is simplest mate.