Quick answer
Stay in central Funchal, ideally between the old town (Zona Velha) and the Lido promenade. Everything you need is walkable, the island's bus network radiates from Funchal, taxis are plentiful, and every tour operator picks up from the main hotel strip. Caniço is the budget alternative, with a frequent bus into Funchal.
Next step
Our picks for these areas are on the Where to Stay page, with price levels and the trade-offs of each.
Context
Funchal is the only base on Madeira that works properly without a car. The old town, the Mercado dos Lavradores, the cathedral, the seafront promenade, the cable car to Monte and the marina are all within a 30-minute walk of each other, and the promenade continues flat for about 5 km west to Câmara de Lobos.
From Funchal, Horários do Funchal runs the city network and the regional operators (Rodoeste, SAM, EACL) run intercity lines from the seafront terminals to Machico, Santana, Porto Moniz, Ribeira Brava and Câmara de Lobos. Fares are low, around €2.05 in Funchal and a €2.65 flat intercity fare, but rural services thin out badly in the evening and on Sundays, so treat buses as a way to reach towns, not trailheads.
Caniço and Caniço de Baixo suit a lower budget: 15–20 minutes into Funchal on frequent buses, quieter, with hotels close to the Garajau sea access. Avoid basing yourself in the north or west (Porto Moniz, São Vicente, Calheta, Jardim do Mar) without a car, the bus back to Funchal can be one or two services a day.
For the mountains and levadas, book pick-up tours from your Funchal hotel or use a taxi for a one-way trailhead drop. That combination, Funchal base, buses for towns, tours for trails, is what most car-free visitors end up doing.