Stay in Funchal, the capital, and if car hire's not your thing you can easily get buses from there to the east and west of the island, amazing scenery. Look for bus tour companies out on the street, don't use any that the hotel or travel operator try and throw at you as they're about twice the price.
If you're after absolute peace and quiet I would get the ferry to Porto Santo and stay there for a night, and hire a couple of bikes. Its beach is never busy, and is so long that you'll easily find a quiet spot.
Funchal has a good selection of restaurants and has plenty of activity, but quite laid back at the same time. Be sure to go to the Armazem do Sal restaurant as well.