Cellular providers shouldn't charge a fee to change cell phones. All you are doing is deactivating your older phone and activating the newer one so it is linked to your account.
Call the cellular provider and reach their customer service. (Usually technical support can do this, but billing and accounts can do it too.) To do this, just dial 611 from either handset to reach the cellular network's customer service.
Ask them to update the register phone on the account to another one you acquired. You will be asked to remove the battery and read them back the IMEI (serial number), then turn off the phone and on again and you should be on their network.
Verizon users would ideally press *228 to program the phone once it's authorized on their account.