As Decatur said.
You can go to court where the trial took place and they'll give you a simple form to fill out. If I remember correctly, it's just a sheet asking for your general information and a section asking your reasoning for filing an appeal. There might also be something about transcripts and a small down payment.
They'll send you off to the nearest court, papers in hand, where your appeal will be heard and you'll finish filing there.
You'll need to pay off your fine first before you can appeal and then you'll wait on your transcripts. This process will take several months. You'll then pick up your transcripts from the court where the trial was originally heard and pay your outstanding balance. The cost will depend on how long the transcripts are, but i'd be prepared to pay at least $100.