How do I access the WordPress files to customize pages and change the layout? I was given a username and password which gives me access to the add text, but not the actual code.
FTP is the default and most popular method. You will need FTP access to the server itself for this method.
Without FTP access you can actually edit the theme code from the WP Admin with the right permissions,
/wp-admin/theme-editor.php. Your best bet is probably to create a child theme and upload it via the theme installer
/wp-admin/theme-install.php this gives you full control but keeps all of the original code in place so updates can be applied cleanly etc.