I would recommend either Cavendish, North Rustico or Charlottetown for your family. Cavendish and North Rustico would have the beach right at these villages as well as offering plenty of great ceilidhs and festivals in the area for live music. There are also a number of great activities including deep-sea fishing, kayaking, parasailing, national park and theme parks all within a short drive. Your teenagers may also be interested in some of the Authentic PEI Experiences found across the Island!
Charlottetown is our capital and there never seems to be an end to the live music possibilities to be found there in the summer time! From Victoria Row to the Confederation Centre of the Arts to the Confederation Landing Park, the whole city seems alive with music. There wouldn't really be a beach within the city, but you would have both red and white sand beaches available within a 20-30 minute drive from there.
Enjoy exploring Prince Edward Island!