I'd recommend Funchal, there are other places to stay, but Funchal has a good offer of decent hotels, plenty of restaurants, shops, bars etc and has excellent transport links.
Plenty to see, some can be reached by bus, but a car offers more flexibility - Curral das Freiras (Nuns Valley), Cabo Girao some of the highest sea cliffs in the World, Pico Ruivo (highest point of the island), Levada Walks, Monte, botanical gardens.
The only really touristy thing we didn't do was the sled ride from Monte to Funchal.