How do I delete text messages ?

In order to delete messages/calls, please follow these instructions:
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Android:

On Talkatone, tap on the Messages icon, select the contact, once the chat log is displayed, tap the Android menu button, and tap on Clear messages/calls. 

If you wish to delete individual messages, just hold your finger over the message you wish to delete for a few seconds. A red button should appear giving you the option to delete. 

Video Tutorial 
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iOS:

Under messages, swipe left on a contact to delete all conversations with that contact, or open the conversation and swipe left to select individual messages to delete.

NOTE: Once you delete the messages/calls log, you won't be able to retrieve this information.