Welcome to Sydney.
I recommend you have a look at Iron Cove Bay which is in the Five Dock/Drummoyne area, so it is not far from your home.
The Dobroyd Aquatic Club has a sailboard division if you are into long boards and they are keen for new recruits. A nice sailing club in a great location. A good spot to learn as you are surrounded by land on 3 of the 4 sides. They do have a Saturday morning beginners/introduction to sailboarding thing available. Recommend this if you are a real beginner.
If you prefer short boards, the Kyeemah Airport runway area would be OK but I did learn short boards at Iron Cove Bay, as you could not be blown out to sea in any wind.
Good luck.