In the pic below I'd very much like to change the words Continue Reading to some other words, like maybe Post Response or some other phrase I'd like to use instead. Is this possible or is it hard coded? I've looked in several files and have been unable to locate where to change that particular phrase.
That's defined in your language file. Look in the lang folder, and find the language file you're using. Assuming it's english, it's named serendipity_lang_en.inc.php.
Look for the define for 'VIEW_EXTENDED_ENTRY'. Change the verbage before the "%s" to be whatever you prefer.
