What is the first file wordpress looks at in a theme?

I’ve inherited a theme and am trying to get my head around a plugin failure.

There is an index.php file in the root of the theme, is that the first file wordpress will look at when it attempts to resolve the homepage or maybe it is something else?

All files in the the root are as follows:

404.php
archive-jetpack-testimonial.php
archive.php
assets
comments.php
footer.php
front-page.php
functions.php
header-front.php
header.php
inc
index.php
page.php
rtl.css
screenshot.png
search.php
secure.php
sidebar.php
single.php
style.css
style.css.map
template--standard.php

With which file does wordpress begin?

If it’s the home page, WordPress will look for the following files in this order and use the first one it finds:

1. front-page.php
2. home.php
3. index.php

WordPress template hierarchy is described in detail here:

Template Hierarchy

This article explains how WordPress determines which template file(s) to use on individual pages. If you want to customize an existing WordPress theme it will help you decide which template file needs to be edited.