The simplest way is to just forward them to your email and then save the email to the computer.
I have saved my email address (the one I use for this) onto my phone, in its own contact listing. Then when I want to save a text just hit forward and then choose that name as to where to send it to, and send it like you would any other message.
DO NOTE: There is a bit of an issue with emails. They usually don't accept multi text messages. So if you are sent a text that is 2 messages long, you will have to split it up to email it to yourself.
Unfortunately phones are not set up to allow for just saving texts as a simple file. I wish they were.
Now you do have the option of using some programs/apps out there that will copy them for you, but personally I don't want anyone else having access to my texts. So I don't really recommend those.