The standard ("ships with Drupal") user page appears to be compacted into the $user_profile() array. Then, it's unpacked and "printed" (to the screen) by the somewhat terse user-profile.tpl.php template:
<div class="profile"<?php print $attributes; ?>>
<?php print render($user_profile); ?>
</div>
Is there a guide somewhere that would help me understand what the typical internal structure of this array is, and what hooks I might need to employ in order to place DIVs, classes and so forth into it, in order to achieve certain basic design goals?
Take a look at https://drupal.stackexchange.com/questions/88407/how-to-create-a-profile-page. I strongly suggest to use the Panels module. It allows you to add blocks, views etc. to your pages and makes theming a lot easier. There are tons of information about Panels on the internet, so I suggest you have a look around first. The following links can be a good start: