How can I save all my text messages?

Text messages, if you are not using iCloud Messages, are saved in your iPhone backup. But they are replaced as you make additions and deletions. If you wish to archive them, you can use a 3rd party program on the computer to download them. TouchCopy is one that is compatible with Windows and Mac, and PhoneView is one that is for Mac only. I know that TouchCopy has a trial version and a paid version. Depending on what you want to do, you can at least use the trial version to see if you like it and if you have a lot of data, you may need to purchase the paid version.

I'm sorry, but it has been a while since I used it. You will need to follow the instructions for the app, then connect the iPhone to the computer. What kind of computer are you using, Windows or Mac? If Windows, is it Windows 10?

EDIT: Glad to hear that you got it working.