Not sure why your phone didn't come with a Google Market or Google Play Store - that baffles me.
But anyway to answer your question, download the Amazon Appstore from their website and then you can download SMS Backup to backup your text messages to a GMail account for archival purposes.
Amazon Appstore: Amazon.com: Appstore for Android
SMS Backup: Amazon.com: SMS Backup+: Appstore for Android
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