It would never do the island justice to just see one part or stay in one place. It's too beautiful.
If you don't want to drive yourself, you can always take a bus tour. La Palma has some top class guided tours. English speaking guides will provide expert information on the surrounding areas and (more importantly) you won't have to concentrate on the road - just the scenery!
Bus tours around the island travel at a slow leisurely pace, giving you plenty of time to absorb all the sights and sounds.
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