Is there any information I need if my phone is lost or stolen that I need to keep somewhere else?

Apple will not help with lost or stolen phones.

Good idea to have the Model Number, Serial Number and IMEI number to report to the police.

Read through this Apple Support document to get an idea of the process in the event of a lost or stolen phone:

If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support

You will need your Apple ID & Password for tracking and erasing device.

You should have the serial number and model number for Police Report.