You've got Our Lady of Lourdes High School (private, Catholic), Poughkeepsie Day School (private, it goes from Pre-K all the way up through high school so I suppose it qualifies), and Spackenkill High School (public). I also think Faith Christian (private) and Oakwood Friends (private) start in elementary school and go straight through 12th grade.
One other public high school that a bunch of kids in Poughkeepsie would end up at is Ketcham in Wappinger, and others would likely find themselves at Arlington in Lagrangeville.