How do I add classes to an img tag inside the node?
you could preprocess the content in the template, but it would be a little difficult, or you could do it via attachBehaviors in jQuery.

If you mind the jQuery way you can add this script:

(function($) {
  Drupal.behaviors.classadd = {
    attach: function(){
      $('.node img').addClass('class1 class2 class3');
    }
  }
}


This way you can easily add classes to your node with .node class. even when you load content dinamicly via ajax. the attachBehavior is a useful Drupal technic to attach actions to your content, and it's widely implemented in all the Drupal framework.