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This one appears free, plus gives the option of backing up on a continuing basis http://www.techrepublic.com/blog/smartphones/backup-your-android-sms-mms-and-phone-logs-to-gmail/

If you press the message for a few seconds it should give the option to forward or copy text. As for an app, check out the reviews first. Sometimes people will tell you if you have to pay up.

There are apps available. http://androidforums.com/threads/best-easiest-app-to-transfer-sms-over-from-old-phone.543658/