Will my new iPhone start working when I turn off my old one?

No. You need to transfer both the content and the network information to the new phone. See this support link→Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

You can transfer the network either by moving the SIM card from the old phone to the new, or activating the eSIM in your new phone on your carrier’s network. Contact the carrier to take the eSIM approach.