Contact Verizon and ask to do just that. The easiest way would be to go into a Verizon store. You will most likely need to have the account owner with you to release your number to a new account.
Depending on your credit history, you may have to leave a security deposit for your line.
It sounds as though you want to become independent from the family plan you're currently a part of! I would love to provide more information on the process of assuming liability of your line. http://bit.ly/RI6Tyh Let me know if you have any further questions or concerns. Thanks!
Follow us on Twitter @VZWSupport