Hi Tom,
I would suggest you start by taking a look at this thread, which provides an overview of how theming works in AtoM, some links on how to create a custom theme plugin, and an explanation of how theme plugin files are typically organized. See:
I think the main file you would want to modify to change the homepage would be homeSuccess.php. Even if that is your only change, I recommend creating a custom theme plugin and providing an override template in the plugin, as this will make it easier to upgrade in the future and reapply your customizations.
AtoM pages are made up of partials though - so for example we have separate files used for the header and footer, and the basic 1, 2, and 3-column layouts on pages (which you can see
here in our code repository), with homeSuccess.php comprising the body elements of the homepage (which I believe uses a 2 column layout as its base). If you modified one of these base templates (either directly, or via a theme plugin), those changes would appear elsewhere in AtoM as well.
It is possible to fully customize the landing page, though! A great example of this is the custom theme used by the Fashion Institute of Technology's Special Collections and Archives' AtoM site:
Hope this helps!