Catholic School might be your only choice, if there are any. The two major private schools I know about were Kamehameha Schools and Punaho School. Kamehameha does not take non-native Hawaiians, and unless you are independently wealthy, you can't afford Punaho. Many years back it was $14k for kindergarden--we're talking college tuition-like amounts here. Might be some others but they are all on the other side of the Island (Honolulu), and not convinient from where you will be living.
For places to live...as an E-6, you could do worse than Mililani Mauka (Upper Mililani). You might also look at Ewa Beach and/or Kapolei if you are willing to drive that far. Honestly, though, if you can get on the waiting list early enough, your very best bet might be military housing, especially if you can get into the newer housing.
