How do I start using an external API and PHP for a WordPress.org page?

@Gina, and welcome!

I’m fairly new to the craft too, but I would love to share what I wish I knew from the very beginning.

Install that plug-in (php-everywhere). Write some simple PHP code to
create some HTML to test it on a test page

When it comes to that I think it’s better to just dig into the themes that produce the front-end. So here’s some resources for you to read.

1. Understanding the WordPress themes. Here’s a link to the Theme
   Handbook. This will help you get around any theme, including the
   ones that come with every installation of WordPress;

2. making and editing a child theme. I think editing a child theme
   is easier than creating one from scratch because one can just go back to the
   parent theme and copy an original file if one made a mess in his child
   theme’ code;

3. another thing that I think will be very helpful and will eliminate
   the need of using the php-everywhere plugin is the knowledge of how
   to make a page template. This will allow you to make a blank
   page that you can fill with whatever you want;

4. of Course, if you get through the entire handbook you might want to
   start your own theme and there’s a nice tool for that one too —
   _s gives you a barebones theme that you can upgrade.

Write a simple API call using the GET and Response Code functions from
the Codex pages. Once I get some data successfully returned, I can
then write more PHP code to incorporate the returned data. (I can use
the web pages written in JavaScript as a guide to write the PHP to
display the data.)

Once you have a child theme with your own page template you can write in it whatever you fancy. Some functions are better suited for the function.php file, but that, I think, will come with the practice and the knowledge.

Hope I helped, good luck!