You have 10 days to explore Madeira? That's plenty of time to get to know this island. If you are spending that long here you should should consider spending 6 days in Funchal and 4 around the island. For example two nights in Zino Palace and two nights at Solar da Bica. In Funchal it all depends on your budget, but Savoy Palace, Les Suites, Cliff Bay and Porto Mare are our main suggestions.
The itinerary is based on 10 days in Funchal so if you decide to stay some of the days outside Funchal you need to adapt your location to each of those days of the itinerary. Renting a car is essential to follow this itinerary.