Since you mentioned "Google Play Store", I'm assuming you're talking about a phone with Android as it's Operating System (OS).
You should include this in your question, along with the OS's version, so people could help you better.
General Background:
Texting is very different than email, regarding how the message is created, kept on your phone, sent over the radio and kept at the destination. You need an app that is specifically written to take your texts and copy them to someplace else for safekeeping.
I've used SMS Backup & Restore for at least 5 years and am satisfied with it.
https://play.google.com/store/apps/d...pRestore&hl=en
My settings: The backup is set to happen every Sunday morning at 4 AM and only if the phone is charging AND connected to the WiFi. (If this isn't available, it will show an alert, so I know to start it manually when it is powered and on my WiFi).
I have it set to backup ALL my text messages, but I don't have that many, and I tend to manually delete (in my Message App) the unimportant ones prior to the backup times.
The storage is to my Google Drive that's associated with the core Google Account for my phone.
Notes:
ANY backup app: They can only back up messages that are currently on your phone. Unlike most other information on a computer (or phone), if you delete a text message, there is NO undelete.
SMS B&R backs up messages to a generic XML format, that's not itself easy to read. You'd need some other program to read them (like MS Word). Or find an app that backs up to an easier to read format.
When you find a Txt Backup & Restore app you like, uninstall the others as you don't want them running in parallel.