What is your goal with these messages?
Are they regular messages, text only, or are they Pictures Videos and words?
Are you trying to save the whole thread, all of the messages sent and received, or just specific messages?
Does this phone have service? If it does not have service, are you able to insert your current SIM card and have service? Some older phones use a larger SIM cards and you cannot switch sizes.
Depending on your answers above, this could be very simple. If you just have some regular text only messages and you just want to capture a few key messages, you could screenshot them, which turns them into pictures of the conversation, and pictures are easy to save and move. A picture of the messages would also preserve the conversation structure. If you are talking about picture or video messages, again these are simple to capture by saving them to the device and then transferring them to a new device. If you are trying to save whole text threads, all messages sent and received, something like the SMS Backup app that was recommended would be better. This all comes down to what your goal is with these messages, are they just memories, something for legal or CYA reasons, or something else. Depending on the reason or use can dictate the best actions.
Let us know what you find, or if you've found your solution and we'll offer more advice.