With a WSOD upon deployment of a site, should I first clear the cache, check watchdog logs, or check the Apache logs?
Yes. Because of the variety of possible Drupal stack configurations, it's impossible to say you'll always find your error in an Apache log. However, a WSOD generally means PHP has hit a fatal error and it's configured not to dump sensitive debug info to screen. 

If that's the case, you might not even have a working Drupal app to make use of its tools like cache clearing, watchdog, etc., so it's best to find where those PHP errors are dumped so you can get Drupal into a bootstrapped state. 

Of course, you should know your stack better than anyone else so a cache clear could eliminate the issue if you know the source of the problem. But if you're unsure, Apache logs are a good first start.