Currently, when I attempt to navigate to a different website's URL in my PHP code, my project just sends me to:
http://localhost/mysite/www.desiredURL.com
How do I navigate outside of 'mysite'?
Sounds like you're missing your protocol in your links. You probably have:
<a href="www.desiredURL.com">link</a>
instead of
<a href="http://www.desiredURL.com">link</a>