Moodle user login with external API

Hi all, need a little guidance or where to look for some information to get some things to work.  Basically what I need to happen is when a user logins, they are authentication to an outside website via API using REST.  Once they authenticate, they are taking to their Student page and can see their course progress, completion etc. (from external site)  I believe once I figure out the login and registering piece, the rest I can figure out.

Here is an example PHP code the execute properly: (This is the login)

https://moodle.org/pluginfile.php/71946/mod_forum/post/1286482/logintest.php

Here is an example user create PHP

https://moodle.org/pluginfile.php/71946/mod_forum/post/1286482/create%20user.php