On your wedding day, you sign three Marriage Certificates. One is for your celebrant to keep for 7 years, one is for Births Deaths and Marriages, and the Form 15 (the "pretty" certificate) is for you to keep. This is your only real evidence that you were married on that day, so keep it safe and sound.
You can't use this pretty certificate to change your name... You need the Births Deaths and Marriages issued Certificate to do that (coz that's the proof that your marriage has actually been registered with BDM).
After your wedding, your celebrant has 14 days to register your marriage.
Some celebrants order the BDM Certificate for you on your behalf, or you can order it yourself here.
Either party can change their names, and you can also choose to hyphenate the two surnames (as long as there is a link between the surnames on the Marriage Certificate and the surname you're applying for, then it is usually pretty stress free!)
Once the BDM Marriage Certificate arrives, you take it down to the Roads and Traffic Authority/Services NSW and change your legal name on your drivers licence/proof of identity card. This is where you sign with your new signature, so you might like to practice it at home a few times first!
Once your drivers licence is in your new name, you need to take your BDM issued Marriage Certificate and your new licence/ID to all the places you need to change your name (banks, Medicare, work, Centrelink, doctors surgery etc).
Too easy! :)