First, remove your SIM card. If you've got a Verizon or Sprint iPhone, you should be able to deactivate your phone through the carrier website, or you can call them and have it done over the phone. This next step is a big one: Go to Settings > Messages > iMessage and turn off iMessage. This will stop your messages from going to your old phone even after you have erased and started using a new phone. Lastly, go to Settings > General > Reset > Erase All Content and Settings. There you can erase all of your content, and safely sell your device, or give it to your mom, without worrying about someone going through your messages.