Hmmm, best kept secrets? The best thing about Disney is Disney! Maybe the best kept secret is Fastpasses are essential to not standing in line. Get a book, figure out how fastpasses work & use them, use them, use them. Some people feel it's a hassle but it isn't.
The other things that is essential to me, is get there before rope drop. Getting up early & getting to the park BEFORE rope drop starts the day off right. In your first hour you'll get soooo much done that the rest of the day is completely relaxed.
We've gone to Magic Kingdom before & had all of Fantasyland (except Philarmagic which can be done anytime) & Space Mountain done by 10 a.m., and that is with a 9 a.m. opening. With that all out of the way you can get fastpasses for Splash Mountain & Thunder Canyon and spend the rest of the day leisurely seeing the sights & shows.
As far as hidden attractions, there is no way for anyone else to tell you what you're family will like. Do your research & go where the moment moves you. But being relaxed (by getting lots done before 11 a.m.) is my biggest secret. If the family feels like an early start isn't vacation, then have them do it your way the first day & promise to do it their way the second day. I've made this deal before with lots of people. Trust me, on day three they will all willing get out of bed even at 5 a.m. to get to a 7 a.m. rope drop. It makes that big of a difference.
Such a big difference that when others read this they'll say, "Estherhead be quiet & stop giving away our secret!"