State laws vary, but we are not aware of a state that allows you to drive by yourself, even to school, when you are only in the learner or instructional permit stage of the licensing process.
The permit typically allows you to only drive with a passenger in the front seat who is licensed. In some states the supervising passenger must be of a certain age (i.e. over 21) or have held a license for a certain number of years (i.e. 4 years). With the permit you are learning the rules of the road and how to operate a car so states require a licensed driver to be your front seat passenger so they can correct any driving errors you may make and help you become a better driver.
In many states once you have spent the required time (normally 6 to 12 months) carrying the permit and taking the appropriate driver's educational classes you can take a test and move up to a provisional license where you can drive alone to school or work but still have restrictions such as driving is prohibited during certain hours (night time usually) and the number of passengers you can take is limited. Laws vary greatly regarding those going through the graduated driver's license (GDL) process so you will need to find what your state's laws are.
For example with a Florida learner's permit you cannot drive alone anywhere including to school or work. You can only drive with a licensed driver who is at least 21 years old and occupies the front passenger seat AT ALL TIMES. In FL with a learner's license (permit) you can only drive during daylight hours during the first three months and until 10 p.m. thereafter, ALWAYS with a licensed driver who is at least 21 years old and occupies the front passenger seat.
If you are convicted of a ticket while in the learner's permit stage of the Florida graduated driver's licensing (GDL) process you will not be able to take the driving test for an Intermediate License for one (1) year from the conviction date.
In many states if you are caught driving by yourself with a permit they will cite you for driving without a license which comes with severe penalties. For example, in North Carolina their Division of Motor Vehicles notes it is unlawful for the holder of a limited learner permit, a temporary permit, or a limited provisional license to drive a motor vehicle in violation of the restrictions that apply to the permit or license. Failure to comply with a restriction concerning the time of driving or the presence of a supervising driver in the vehicle constitutes operating a motor vehicle without a license.
In NC driving without license is a Class 2 misdemeanor. A Class 2 misdemeanor in North Carolina usually comes with a fine and/or jail time. It is our understanding that it is up to 6 months in jail and or up to a $500 fine.
To find out what restrictions and conditions you have associated with your learner's permit and if you can drive yourself to school alone and without a supervising licensed passenger, check with your state's Department of Motor Vehicles. Before driving with your permit make sure you are properly insured to drive. CarInsurance.com can help you receive a quote on insurance and plenty more information on car insurance for young drivers.