You’re unable to copy entire messages at once using the default messages app. You can, however, copy one at a time if the conversation isn’t that long.

Repeat this copy and pasting process to copy all the text messages you want.

If you don’t have Textra, it’s a free app that’s highly rated and very popular because it has so many useful features. When you start using Textra, all your previous text messages from your default SMS app will also appear.