Hi there,
Do you want the same links to show in the drop-down Browse menu available in the AtoM header bar?
If yes, then you can customize the Browse menu via Admin > Menus to add the additional links. See:
In terms of customizing the look and feel of the browse options in the home page, it may be useful to look at how the browse menu is rendered in the arArchivesCanada plugin, to see how we've implemented that. See for example:
Here is how it looks on the ArchivesCanada theme homepage:
This previous thread in the user forum includes a lot of introductory information on developing AtoM theme plugins, including how they are organized:
You might try starting your custom theme with the arArchivesCanadaPlugin, and changing the icons found in the images directory, along with other assets and styles. The Corcovado plugin is very simple, created just for demonstration purposes!
Hope that helps!