Teachers!
Meet them. Find out it they are planning to stay. (I got burned on that once when splendid teacher left 2 weeks into the preschool yr. and apparently parents not new to the school like me knew this would happen!)
Sometimes a school has a very charismatic principal or leader, but the teachers are mediocre. Make sure you meet the teachers. An intelligent and motivated bunch of teachers can work around resource issues. And an apathetic group can do little with the tools handed to them.
What are their philosophies on learning and discipline? How do they structure the day? How do they account for different learning styles and different levels of advancement in the same age group.
Also look at class size, no matter how talented a teacher, too many kids to handle can wreck the learning potential of everyone. Also, how does the school handle special needs kids (ADHD, Autism)? Is their enough money for the class to get an additional teacher to compensate for the attention this/these children will need?
I am sure other people will have great advice, but this is what I've learned over the yrs. My kids are only 5, 7, and 9. They went/go to private preschools and go to public elementary school.
Oh, here is another thing I'll add which might come off badly, but try to look at who are the other kids and families that go to the school. In general what are their values and beliefs. Is your child going to be surrounded by children whose parents only buy them expensive clothes and iPods? Is your child going to be surrounded by overworked parents who cannot participate in its events or boost the schools' function?
My preference if possible is for an economic, social and racial mix--so my kids learn more about the world: that some kids will have their own Mac laptops in third grade, and others will wear 4x hand-me-down clothes.