Change URL only on WordPress+CiviCRM
The advice in the documention is fairly old and much of it is no longer appropriate. In the most recent CiviCRM versions, more and more of the url settings are automatically calculated from the 'base url' and are relatively easy to update just by editing the CIVICRM_UF_BASEURL in your civicrm.settings.php file, and then clearing your caches for good measure. In fact, most of the time you don't even have to worry about that, you can access much of your civicrm from multiple domains (e.g. with and without www). Of course, that's not usually a good idea, and the one key time that the settings file matters is when your civicrm is invoked via a non-CMS entry point (e.g. the click-through tracking url from CiviMail).

Usually, it's when you also have to change file locations that it gets the most icky.

The one recent gotcha I've seen is a ckeditor security check for file uploads that will break things in a small way if you don't have the file locations right (e.g. you need the actual file path instead of a symlinked one). But that's a digression for completeness and unrelated to your question....