Hi Barb79,
The appeal process (particularly in the Supreme Court) is extremely complicated. It is a subject in itself at university. Hence, your question needs to be a little bit more specific.
First though, the appellant (person appealing) needs to have the leave (consent) of the higher court for appeal. You do this by lodging an application for leave for appeal. You can appeal against the conviction itself or the sentence:
- This page from the Supreme Court may help you (assuming you are appealing to the Supreme Court of Victoria): "Criminal Appeals"
As for costs, it really varies on the nature of the case. It will be expensive. If you ultimately win the appeal, the State will bear the costs. If you lose, the court will make an order as to costs and the likely outcome is you pay for the state's legal fees and expenses (not actual but a taxed basis) as well as your own.