Keep in mind that if you want to do some SCUBA diving (and I'd strongly recommend that, even if you have no prior experience - it is truly magnificent), you can't fly for 18-24 hours after the dive. That means that you may want to schedule a day or two more around Cairns or Port Douglas.

for a day trip, rent a car and drive north to Cape Tribulation along the Captain Cook Highway, the drive is beautiful.

Going to the Reef, we've used Quicksilver (it goes from Cairns, stops at Port Douglas, then continues to the Reef). If you have no prior experience, they have introductory diving, when a group of 4 beginners dive with an instructor.