I currently stay in Melbourne Australia and I want to get to Byron Bay. So what is a convenient and inexpensive way to get from the Gold Coast Airport to Byron bay? For example: a Shuttle bus service.
Firstly you'll want to keep in mind that the "Gold Coast Airport" is not really on the Gold Coast - it's in Coolangatta (or more correctly, half of it is in Coolangatta, half of it is across the state border in Tweed Heads).
There are a few coach services between the Coolangatta/Tweed Heads areas and Byron Bay as a part of longer runs (eg, Brisbane -> Sydney), such as Greyhound. Although there are no trains to Byron, CountryLink does run a bus service from Tweed Heads. In both cases you'll need to get to wherever the bus picks up yourself - either via Taxi or a local shuttle bus.
There are also private shuttle companies that will go from the airport to Byron Bay, but they are expensive for just one person. eg, Gold Coast Shuttle will charge $75 for up to 2 people, and $20 for each additional person - not a bad price if there's at least 2 or 3 of you, but far more expensive than the other options if you're alone.
Your other option is to fly into Ballina and get a shuttle bus to Byron. eg, Xcede charge ~$15 each way from the "Ballina-Byron" airport (which is in Ballina!) to Byron Bay.