Should I remove the SIM card on an old iPhone I’m trading in?

but am I supposed to put the SIM card from my old iPhone XR into the new iPhone 13 PRO?

Yes, you do.

The new phone came with a little tool to open the SIM card slot, but no instructions as far as what/why I’m I need to do anything with it.

Check here -> Remove or switch the SIM card in your iPhone -> Remove or switch the SIM card in your iPhone - Apple Support

This is the little tool, looks like a paper clip.

Yes, then you can use the same phone number for your new iPhone 13 PRO, if it is an carrierr unlocked iPhone.

I asume you use also AT&T as carrier for your new iPhone 13 PRO.

That is also handy then you don't have to change anything, you use the same number.

IF you use another carrierr for your new iPhone 13 PRO, you can still use the same phone number.

Let us know if that is the case.