You should definitely remove the SIM card. That's your number linked to it, unless you can get Verizon to un-link it. If you've saved any data to it (Contacts, usually), then you need to remove them from the SIM card storage as well.
If you're to sell your phone, make sure you wipe it clean (factory reset) or any data or information you've set it up with will still be there up for grabs. If you factory reset the device before you sell it, then the new user wouldn't have to do anything else, just set it up with their account.