I realize that there is a lot more global interest in the British Royal Family (even in this thread, the title and the vast majority of posts focus on them), but the Swedish royal family is a good example of combining royal life and family life successfully - I believe it's come up in this thread before. They release pictures of the kids on occasion, but not too often; the kids have some official appearances (especially Princess Estelle) but also plenty of time to be kids. For each child, there was a Te Deum service and the announcement of the name within days, and a public christening a few months later, with official photos and information released; here's
the most recent example, and note that the father is an untitled private citizen.