What kind of license can you get if you're under 16, and have your state permit?
You usually can only get it if you have a worker's permit. It allows you to drive on you're own if you're 15, if you're going to either school or work, and if you have your own state permit.
Driving laws and rules for the issuance of licenses are taken care of by the states, so it depends which state you live in. In many of them, you cannot obtain any kind of license before you are 16. Some of the more rural, farm-based states, however have an agricultural license, meaning you can drive vehicles and farm equipment as long as you don't do so off of private property on actual public roads.
Idaho, for example, will allow an agricultural license if you are 14 or 15 years old.
All that a permit does is allow you to practice driving under the supervision of an adult, and usually one with a minimum number of years experience as a driver, and even when you receive your actual license at 16, there are often restrictions as to who and how many people may travel in the car with you.