Homeschool Laws Vary by State
Here are examples of states varying between very lax homeschooling regulation to highly regulated. I am comparing Utah and Texas, which are both relaxed, to Washington, which is considered moderate, and Pennsylvania, which has high regulation. You can easily see the variance.
Several states, Utah included, do not have rigorous homeschool laws. In Utah, for example, you need to teach certain subjects for a certain length of time, and you need to provide a signed affidavit saying you will do this. However, there is no oversight for this. There are no requirements for assessment, and the state has very few rights to ask whether or not you are following these rules. They cannot inspect where you teach, or decide whether or not you are qualified to teach. As soon as you provide the affidavit, you will be allowed to homeschool in Utah.
If that sounds too uptight for you, groups of families may form a "private school" that is exempt from regulation.
In Texas, homeschool families are also highly protected. Similar to Utah in the amount (or lack of) regulation, Texas does go further than Utah and actually protect homeschool families' rights to choose their method of teaching their children.
The subjects covered need to include: