Everyone has his favorite, talking about levadawalks and explorings at Madeira. I have walked some of them through more than 13 years. Here is two of my favorites - totally different from each other, but both with a spesific beauty you will remember for ever. Both my favorites you can walk without guid, with good shoes and a bottle with water. BUT: Take care, take time, take the silence.
For the preparence. Buy the book: Madeira car tours and walks, of: John & Pat Underwood. My reference is according to this book.

Madeira is the forgotten island for the fourthies, the paradise for the sixties and the future for the twenties. The changes the last five years is nearly unbeleavable. The rapid-road-system is excellent developed. Beatiful ocean bathlands has been build all over the island. The hotel-city at Lido, outside Funchchal, only 20 minutes with taxi from the airport, has all the hotels, stores, restaurants and activities you want.