A custom Genesis blog page displaying three main sections. The 1st section is a featured post from category #1. The 2nd section shows some meeting info. The 3rd section displays posts from all categories except the post displayed in the 1st section.

home.php

<?php

// Add sections above the main blog loop
add_action( 'genesis_before_loop', 'bds_featured_post' );
add_action( 'genesis_before_loop', 'bds_meeting_info' );

// Replace the blog loop with custom loop
remove_action( 'genesis_loop', 'genesis_do_loop' );
add_action( 'genesis_loop', 'bds_do_custom_loop' );


// Run the Genesis loop.
genesis();


/* ========== Build Functions ========== */

// Following instructions for excluding a post already displayed: https://codex.wordpress.org/The_Loop#Multiple_Loops_in_Action

// Build featured post section
function bds_featured_post() {
  $featured_posts_query = new WP_Query( 'cat=1&posts_per_page=1' );

  // Start the loop
  if ( $featured_posts_query -> have_posts() ) :
    echo '<div class="latest-update-wrapper">';

      do_action( 'genesis_before_while' );
      while ( $featured_posts_query -> have_posts() ) : $featured_posts_query -> the_post();

        // Create a global variable to use later in the bds_do_custom_loop()
        global $exclude_post_id;
        $exclude_post_id = get_the_id();

        do_action( 'genesis_before_entry' );

        genesis_markup( array(
          'open'    => '<article %s>',
          'context' => 'entry',
        ) );

          do_action( 'genesis_entry_header' );

          do_action( 'genesis_before_entry_content' );

          printf( '<div %s>', genesis_attr( 'entry-content' ) );
          do_action( 'genesis_entry_content' );
          echo '</div>';

          do_action( 'genesis_after_entry_content' );

          do_action( 'genesis_entry_footer' );

        genesis_markup( array(
          'close'   => '</article>',
          'context' => 'entry',
        ) );

        do_action( 'genesis_after_entry' );

      endwhile;
      do_action( 'genesis_after_while' );

  else : // If no posts exist.
		do_action( 'genesis_loop_else' );

    echo '</div>';

	endif; // End loop.

  wp_reset_postdata();
}

// Build meeting information section
function bds_meeting_info() {echo "Coming Soon";}

// Build custom blog loop that excludes the post from the featured post section
function bds_do_custom_loop() {

  // Get post ID from the post displayed in the Featured Post section
  global $exclude_post_id;

  // Start the loop
  if ( have_posts() ) :

    do_action( 'genesis_before_while' );
    while ( have_posts() ) : the_post();

      // Exclude the post from the Featured Post section and continue the loop
      if ( get_the_id() == $exclude_post_id ) continue;

      do_action( 'genesis_before_entry' );

      genesis_markup( array(
        'open'    => '<article %s>',
        'context' => 'entry',
      ) );

        do_action( 'genesis_entry_header' );

        do_action( 'genesis_before_entry_content' );

        printf( '<div %s>', genesis_attr( 'entry-content' ) );
        do_action( 'genesis_entry_content' );
        echo '</div>';

        do_action( 'genesis_after_entry_content' );

        do_action( 'genesis_entry_footer' );

      genesis_markup( array(
        'close'   => '</article>',
        'context' => 'entry',
      ) );

      do_action( 'genesis_after_entry' );

    endwhile;
    do_action( 'genesis_after_while' );

  else : // If no posts exist.
		do_action( 'genesis_loop_else' );

	endif; // End loop.

  wp_reset_postdata();
}

Thank you for the tips! Very helpful!

Here's what I ended up doing...

I couldn't use offset because I wanted pagination and I didn't want to exclude just any post. I wanted to be sure that I was excluding the post already displayed in the upper section. Final product: http://brubakerministries.org/updates/

// Remove header
remove_action( 'genesis_header', 'genesis_header_markup_open', 5 );
remove_action( 'genesis_header', 'genesis_do_header' );
remove_action( 'genesis_header', 'genesis_header_markup_close', 15 );

// Replace the blog loop with custom loop
remove_action( 'genesis_loop', 'genesis_do_loop' );
add_action( 'genesis_loop', 'bds_do_custom_loop' );


// Run the Genesis loop.
genesis();


/* ========== Build Loop Functions ========== */

// Build custom blog loop that excludes the post from the featured post section
function bds_do_custom_loop() {

  // Only show these sections if displaying page #1
  if ( ! is_paged() ) {
    // Add sections above the main blog loop
    bds_featured_post();
    bds_meeting_info();
  }

  // Run a query to get the ID of the latest featured post
  $exclude_posts_query = new WP_Query( 'cat=1&posts_per_page=1' );
  if ( $exclude_posts_query -> have_posts() ) {
    while ( $exclude_posts_query -> have_posts() ) {
      $exclude_posts_query -> the_post();
      $exclude_post_id = get_the_id();
    }
  }
  wp_reset_postdata();

  // Setup the custom loop
  global $paged;

  $args = array(
    'post_type' => 'post',
    'post__not_in'  => array( $exclude_post_id ),
    'posts_per_page'=> get_option('posts_per_page'),
    'paged'         => $paged,
  );

    // Get correct featured image size for custem blog loop section
    add_filter( 'genesis_pre_get_option_image_size', 'bds_blog_posts_image_size' );

    // Customize the entry meta in the entry header
    add_filter( 'genesis_post_info', 'bds_blog_posts_entry_meta' );
    function bds_blog_posts_entry_meta($post_info) {
      $post_info = '[post_date] [post_comments]';
      return $post_info;
    }

  // Output the custom loop
  echo '<div class="blog-posts-wrapper">';
    genesis_custom_loop( $args );
  echo '</div>';

}