From tasting the island’s namesake wines to dining in Michelin-starred venues, Madeira now attracts food-obsessed travelers who once skipped the island for the mainland. Here’s why that’s changed.

The island, a Portuguese territory, is only 35 miles long, but its mountains are laced with scenic trails, tumbling cascades and inviting villages. The Cascata do Risco lies at the end of the Levada ...