Cairns is a small, hot, easygoing city that exists because of what surrounds it. It was founded in 1876 as a port for the Hodgkinson River goldfields up on the tablelands, and today it is the place where two World Heritage areas meet: the Great Barrier Reef offshore and the Wet Tropics rainforest behind it, which includes Daintree National Park. The town itself is short on sights. The Esplanade, a free saltwater lagoon, a market and a handful of pubs is most of it, and that is fine, because you are here for the edges.
Five days is the right length because the highlights are all full days and none of them overlaps. The reef is a day on a boat. Daintree and Cape Tribulation is a long day on the road. Kuranda is a day of cable car and railway. The Atherton Tablelands waterfalls are another long loop. Fewer days and you drop one of them; more and you start repeating yourself. This plan also deals honestly with three things the brochures skip: the reef has been through serious bleaching in recent years, November is the start of marine stinger season, and the Daintree has crocodiles and cassowaries that are not decorations.
Day one is Cairns itself, a gentle start. Day two is the Great Barrier Reef, with a plan for choosing between the outer reef and Green or Fitzroy Island. Day three is the Daintree, Mossman Gorge, the cable ferry and Cape Tribulation, with dinner in Port Douglas. Day four is Kuranda, up by Skyrail and down by railway. Day five is the Atherton Tablelands. If you want more of the reef, the Whitsundays are a different, southern version of it, and the Australia page puts both in context. Start on a Friday: Rusty's Markets only runs Friday to Sunday, the museum and Ochre are closed on Sundays, and the reef day lands on a Saturday when tours run at full strength.