Yes, there is an app called "Email My Texts SMS" that will do just that. Email all your texts or a thread of texts to yourself and then print them out. You can even save the email as a backup.
I hope that helps.
It would depend on your phone. Since you posted this in the Basic section, I would assume it's a basic phone, and therefore, downloading an "app" would not be a workable solution.
Short of emailing the messages one at a time, the only other option I know of for basic phones is a third party app called BitPim. You can download all the texts saved in the phone's memory to an Excel sheet and print them out from there. Not all phones are supported by BitPim, and if you don't follow directions well, you could brick your phone and void the waranty, so use it with caution.
Check out the application I posted at https://community.verizonwireless.com/thread/770529, it works on basic device and smartphones well, the full version is a bit costly but it allows you to use the app for free a number of times before it request you buy, I bought the app because it backup everything I keep in one area and at one time..
Give it try it may be the one for you..