How do I get my notes back after getting a new iPhone

Assuming you synced notes with iCloud, on your new iPhone, open the Settings App. Tap your name at the top, then tap iCloud. Tap Show All and turn on all the switches for things you likely synced with iCloud and if you did sync them, they will sync to your new iPhone. This does require using the same Apple ID on the new iPhone as was used on the old phone.

it should if you have restore it from a backup of the old or if you have set it to sync noted with icloud