using jQuery Cycle by Malsup with WordPress Page and Gallery

What your asking for has the basic functionality of a plugin, so saying you don’t want to use a plugin doesn’t make much sense.

If you do for some reason want to write your own , using any jquery or JavaScript is relatively straightforward, you enqueue the script where you want it, then wrap whatever data your using in the javascript tags ( usually a div class or id ). This is a very basic example.
http://codex.wordpress.org/Function_Reference/wp_enqueue_script

 wp_register_script( 'jquery_cycle_name','http://path-to-your-cycle.js');
 wp_enqueue_script( 'jquery_cycle_name' );

The javascript:

jQuery(document).ready(function($) {
    $('#featured-works').cycle('fade');
});

Your template or whatever you wrap your info in, typically you want this in a loop to go through several images/posts or content.

<div class="featured-work"> images, etc, go here</div>