Which PHP functions would I use to load pages of a WordPress site in conjunction with AJAX/Infinite Scroll?

Which PHP functions would I use to load pages of a WordPress site in
conjunction with AJAX/Infinite Scroll?

I'm new to PHP. I'm building a custom WordPress theme for a music website
that sells 3-album bundles and am trying to integrate this:
-User can see the previous week's bundle by going to the 'Last Week's
Bundle" page. From there, there is also a sorting feature that will allow
them to either sort bundles by most recent, popularity, alphabetically, or
just shuffle them.
On the front-end, my colleague is integrating some AJAX that will allow
the content to be served dynamically without the page reloading, as well
as an infinite scrolling feature.
Being new to PHP, I am trying to learn which way I should go about serving
content to the browser as the AJAX requests it. Looking in the WordPress
Codex, I cannot find enough about returning only a certain number of posts
and how to make the loop grab the next set of posts when the user hits the
'more' button, as well as having it sort all at the same time.
Here's my code below:
<?php
$sort = $_REQUEST['name'];
if($sort = "recent") {
$query = new WP_Query( array ( 'orderby' => 'date', 'order' => 'ASC',
'post_count' => 3 ) );
if (have_posts()) : while($query->have_posts()) : $query->the_post();
}
if($sort = "alphabetize") {
$query2 = new WP_Query( array ( 'orderby' => 'name', 'order' => 'ASC',
'post_count' => 3 ) );
if (have_posts()) : while($query2->have_posts()) : $query2->the_post();
}
if($sort = "shuffle") {
$query3 = new WP_Query( array ( 'orderby' => 'rand', 'post_count' => 3
) );
if (have_posts()) : while($query3->have_posts()) : $query3->the_post();
}
?>
Also, here's a snapshot of the front-end:
http://d.pr/i/lb0G/1piqyzsc