Roberstons barracks are near Palmerston from memory.
The best school is closer to Darwin, being Essington. It goes start to 10 I believe.

Having went to school in a few darwin primary schools and only just having left Darwin, I was only prepared to put my girls in there.
I don't know if its because not being a state, theres no state schools. But I found most public schools to be quite lax, especially in the palmerston area. I feel horrible for saying it, but its my opinion on it. Friends of mine attended Essington and swear by it. And when we first moved up it was the most recommended school.

Essington I believe to be public (cant be sure) but it is the best school in Darwin.

HTH