I live in Louisiana, and the laws here are pretty simple.
You have 2 choices...you can declare your own home a private school. This means sending a letter to the State Board of Education before the start of every school year stating how many days your students will be attending school, and how many students you have. You have to choose a name for your school, too. The law protects your privacy, and the privacy of your students.
The other option is to have an at-home-study-program that works with your neighborhood school...regular testing at the school, and you are required to have an accredited teacher do the instruction.

The private school in your home is the way to go. Their only requirements are that you teach the Pledge of Allegiance, and the Constitution...everything else is up to you.

It's great here in New Orleans because private school is insanely expensive, and usually includes some form of religious study. And, the public schools are sub-par.