Need help getting recent text & imessages into my mac under messages.

I have texts from 2014 in Messages on my mac, but no recent ones. I'd like to save certain texts for work. I've recently upgraded my iphone and now would like to make sure I've saved the old texts on my computer. I've backed up my old iphone to iTunes. Now how do I get those texts into Messages and how do I get new ones from now on (on my new iphone) to be stored there as well?