How do you get notes to stay only on your iPhone and not sync to any other device on the Notes app?

Notes may sync via other email services, like gmail.

You need to check the settings of EACH of your email accounts, and, if necessary, disable syncing notes.

It is also possible that notes sync to your Mac when you connect your iPhone to iTunes (via the USB cable or wireless). To avoid this, you might want to check iTunes sync settings (not 100% sure about the latter is still valid, it was definitely present in older versions of iTunes).

Simply go to Settings > Notes and make sure the default notes account is set to "On My iPhone" instead of "iCloud".

Now, newly created notes on your iPhone will remain on your iPhone and will not get pushed to the iCloud and hence your iPad.

Go to the iCloud settings on your iPhone [Settings/iCloud] and simply disable the Note's sync features.