By plane
The Gold Coast Airport IATA: OOL (http://www.goldcoastairpo...), also known as Coolangatta Airport, is in Coolangatta to the south. It may be unique in the world in that, because it straddles the NSW and Queensland borders, you land in one state and arrive in another.
It is a fairly small terminal but handles around 3.5 million passengers per year with frequent connections from major Australian cities and some international flights from New Zealand and Asia. It is NOT a 24hr airport and closes at night, so don't plan on spending the night before an early morning flight. It is around 30 minutes drive to Surfers Paradise. There is a shuttle bus connecting the airport to the main Surfside buses route on Gold Coast Hwy between Tweed Heads and Surfers Paradise.
A viable alternative, especially if hiring a car, is to fly into Brisbane Airport IATA: BNE. You can catch a direct train from the airport 105 minutes to Nerang 8km to the west of Surfers Paradise, and connect with the local buses. Otherwise it is a 90 minute drive down the Pacific and Gold Coast Highways.
By train
Queensland Rail runs an electric CityTrain service (http://www.citytrain.com.au/) from Brisbane to Coomera, Helensvale, Nerang and Robina. Trains run half hourly from 6 am to midnight, 7 days per week taking approximately 70 minutes from Brisbane Central to Nerang and Robina. The trains can get crowded at peak times. Buses connect from Helensvale, Nerang and Robina to Surfers Paradise, Coolangatta, and into northern New South Wales.
As the train runs parallel to the coast you should check which station is actually closest to your final destination in the Gold Coast.
Countrylink in NSW (http://www.countrylink.info) runs trains from Sydney with bus connections at Casino for services direct to Surfers Paradise or Tweed Heads. However this is a very slow, 11.5 hrs, and expensive service. Set aside a whole day for the trip.