How do I keep my text messages from showing up on other iPhones and iPads?

Be certain each and every person, regardless of age, has her/his own, unique Apple ID.

Hopefully you have only one iPhone using your Apple ID, so it will be the only one with your messages. Regarding your iPad, simply don’t do text message forwarding.

How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac - Apple Support

If you’re currently in a situation involving shared Apple IDs, extensive cleaning will be required to undo this setup.

Each person acquiring a new Apple ID must do this to her/his device(s) before “moving in”:

What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch - Apple Support