How do I keep texts from appearing on both my iPhone and iPad?

Do you mean SMS messages (green) or iMessages (blue) or both? If you mean iMessages or both, go to Settings>Messages and sign out of iMessages. This will prevent you from getting iMessages and any SMS texts forwarded from your iPhone.

If you still want to get iMessages but not SMS messages on your iPad, on your iPhone, go to Settings>Messages>Text Message Forwarding and remove your iPad.

When you turn on Messages in iCloud, every message you send and receive on your iPad is saved in iCloud. And, when you sign in with the same Apple ID on a new device that also has Messages in iCloud turned on, all your conversations show up there automatically.

On your iPhone > Go to Settings  > [your name] > iCloud, then turn off Messages to stop displaying messages on other devices