Hi Bart, based on what you say, I would get up to http://www.capetribbeach.com.au/ for two maybe three nights of your time, there being tours for around $100 that do a split by taking half a day or so to take you up there and then pick you up on whatever day you plan to leave, usually around 2 pm. I think it was from memory.
There is also a public bus service - Coral Coast it may run under.
A great rainforest experience there with walks/climb of Mt. Sorrow if you are game to battle a jungle environment with leaches in a few places so make sure of getting some salt into your boots and socks - plenty of it if you intend to do that climb for some great views out over the reef from rocky outcrop way up the top if no clouds obscuring the view. Also board walk and night walks you can do there.
There is a free great walk out of Cairns itself, just follow the train line on western side and head up twards the Botanical Gardens and you will go through a wetland rainforest environment on boardwalks and then adjacent to the Botanical Gardens there is a Gwondaland Interpretive Forest Walk, a Climb up to a look out over Cairns and also some adapted old WW2 storage tanks worth having a peek at.
Plenty of reef trips out of Cairns so take a stroll along the Esplanade on arrival and see what deals are on offer, it cheaper to take a slower ( sometimes even a bit of sailing ) boat to inner reef ( islands with reef ) locations or the faster big Cat for a 1.5 - 2.0 trip to outer reef, probably up around $200 these days.