How can I find out if my new iphone is refurbished

There is no sure way if the phone came from a non-approved source. But if you bought it directly from Apple or from an Apple authorized reseller it is not refurbished unless you specifically asked for a refurbished phone to save a little money. If the Model Number begins with M or P the phone is original. If it begins with N it was refurbished by Apple. If it begins with F it was refurbished by someone other than Apple.

You can also enter the serial number here to verify that it is actually new—>Check Your Service and Support Coverage - Apple Support

You can't. The Serial number does not identify if its refurbished or not.

You must use the model / part number that looks like MXXXxx/X in Settings->General->About->Model

If it starts with an "N" or a "F" its refurbished.

If it starts with an M or a P it was a new retail unit (engraved in the case of P).

3rd parties cannot change the model number like that. They don't have access to it. 3rd party refurbishments retain the M or P

F or N was refurbished by Apple. It's just defines how it was provided by Apple.

F for purchased from the refurbished store.

N if it was used as a replacement iPhone for a service request.

Both items would have been refurbished. Though the N models can be new units set aside for service replacements as well.

What is the first letter of the serial?

Click this link: https://osxdaily.com/2018/01/27/determine-iphone-new-refurbished-replaced/