You can start by catching a 615 bus from Mooloolaba to to Landsborough that takes about 30 minutes.
Queensland rail have a station at Landsborough on the Sunshine Coast Line that runs into Brisbane with final stops at Bowen Hills, Fortitude Valley, Brisbane Central and Roma Street.
The Gold Coast line starts at Bowen Hills, also with stops at Fortitude Valley, Brisbane Central and Roma Street, and then onto Varsity Lakes, the stop in Mudgeeraba.
I assume you will be over-nighting on the Gold coast. Googling "How to get from Mudgeeraba to the gold coast airport" lists a variety of options. A shuttle service is probably the most expensive but may be the most reliable for an early morning flight.
Bus route 760 runs from Robina (the Gold Coast Line stop before Varsity lakes) and can take you on to Coolangatta in around 45 minutes (one of your options). The stop before Coolangatta is the airport!
A quick estimate put the cost of Mooloolaba to Coolangatta transport at around AUD$50 or so. If you already have a go card (plastic transport card) and don't need a paper ticket it may be even cheaper.