You can bring your own bike with you on the plane, of course, but you have to take them apart to fit into those cardboard boxes. What I prefer to do, whether I'm in Ireland or Provence, is to pack a folding bike rack in my suitcase, then put it on the back of a rented car, rent bikes in the first town I come to and take them with me. Then on rainy days I drive; on beautiful, sunny days, I bike. You can go on day trips, of course, but you can always arrange with your hotel to bring your bags by taxi to another town.
If you don't like rubbing saddlebags with strangers, you can usually convince bike outfitters to customize an itinerary just for you and your family, with bike pickups and deliveries. If you want to travel with a group and can afford the price, I recommend Backroads at 800/462-2848 or www.backroads.com. I like it that the day ends with a hot bath in a hotel soaked in character and charm. And I like it that everyone gets their own route map, so you can go at your own pace and be as sociable, or as unsociable, as your heart desires.