It is approximately 25-30km (or about a 20 minute drive) Cairns is located on Trinity Bay, that's where all the name references to Trinity come from. There is a map here http://www.visitcairns.com.au
From the Sunshine Coast (say, Maroochydore) to Cairns is a distance of 1596 km. It would take about 20 hours of continuous travel, so the journey is best undertaken over three days, allowing for rest...