I want to take text messages from my iPhone and save them on my computer. Is it possible to save iMessage conversations to my PC?"
Each time you backup your iPhone, text messages are stored by iTunes as part of the iPhone backup, just in case you need to reset your iPhone or restore from backup. These backups aren't searchable, so you can't easily view them on your computer. If you use secure backups, which you should to protect your personal information, the iPhone backup is also encrypted. Thankfully, you can backup your phone for recovery purposes and use a second software app to copy all those SMS text messages from iMessage on your iPhone and transfer them to your computer where you can search and archive the messages however you see fit.
The software I like for copying text messages from your iPhone is Tansee iPhone Transfer SMS.
Tansee iPhone Transfer SMS imports all the iMessage conversations on your iPhone into a searchable file on your computer. You can password protect this backup, so that no one else can view the files if you share your computer.
The software includes a built in text message reader, so you can easily browse the messages and read the text from the convenience of your computer. The interface is incredibly simple to use - you basically just connect your iPhone to your computer and click a single button to copy text messages to your computer.
Here's a video showing you how the process works, so you can see how easy it is to backup your iPhone text messages to your PC: