Yes, there is an easy way, but it depends on the application you're using. Most applications allow you to save the conversation on your phone or email it directly to your email. Open the application you use and then look for Share or the paper clip icon. Choose your email and fill the details. The attachment will contain the text messages, but the pictures need to be saved automatically from the Gallery by connecting the phone to your computer. This procedure works for most of the chat and messaging applications.
