Madeira is a volcanic island in the Atlantic, about 805 km southwest of mainland Portugal, and it is steep in a way photos don't prepare you for. Portuguese sailors claimed it in 1419 and settled it a few years later, and the settlers did something remarkable: they cut more than 2,000 km of irrigation channels, the levadas, along the mountainsides to carry water from the wet north to the dry south. Today those channels are the best walking paths on the island. Over the centuries the island's money came from sugar, then wine, then English and German visitors escaping winter, and the last of those never really left.
It is not a sandy-beach island, and anyone who tells you otherwise is selling a package. Funchal has a harbor and a seafront with pools, the natural lava pools in the north are fun, and the golden beach is on Porto Santo, a separate island. What Madeira does offer is mountains you can drive up to 1,800 m, a laurel forest that is a World Heritage Site, cliffs that drop 580 m into the sea, a food scene built on grilled beef, limpets and a very strong lime drink, and a climate that is mild all year. Five days is the right length because the island looks small on a map and drives like a large one: every outing is an hour or more each way on bendy roads.
The plan keeps each day in one part of the island. Day one is Funchal itself, its market, its old town and the hill of Monte. Day two is the mountains, with the Pico do Arieiro to Pico Ruivo crossing. Day three is the north-east, a levada walk and Santana. Day four is the long loop through the west and north coast, and day five is the east and a wine lodge. For the mainland side of the country, see Portugal. Start on a Monday if you can: the market and the toboggan are both closed on Sundays, and the market closes at 2pm on Saturdays.