I live in NY STATE and I have a jr drivers license which means I can drive in the state alone (but with restrictions) do I have to be put under my family's insurance or am I automatically covered?. meaning can I drove alone right when I get my license?

In NY one insures a car, not entirely a driver. However, if you will be driving one of your family's cars regularly, then they should list you as an authorized driver.

You will have to be "listed" as a driver on their car if it is a family member living in the same household. Since a teen driver will cost double or triple the normal insurance rate of an adult normally.
So you don't get a separate policy but you do get listed as a driver on them.