Show parent’s child and also child’s, child on a page?

Will get_pages() work for you? http://codex.wordpress.org/Function_Reference/get_pages

If I’m understanding your question correctly, you would want something like this:

$children_of_page = get_pages(array(
    'child_of' => $post->ID,
    'parent' => $post->ID  //defining both parent and child forces the list to only include direct children.
));
$children_of_first_child = get_pages(array(
    'child_of' => $children_of_page[0][ID],
    'parent' => $children_of_page[0][ID]
));

If you wanted to output a list directly, you could use wp_list_pages(), http://codex.wordpress.org/Template_Tags/wp_list_pages, something like:

wp_list_pages(array( //Get first level children of current page and output as list
    'child_of' => $post->ID,
    'depth' => 1,
    'title_li' => ''
));
$children = get_pages(array(
    'child_of' => $post->ID,
    'parent' => $post->ID
));
wp_list_pages(array( //output the children of the first child
    'child_of' => $children[0][ID],
    'depth' => 1,
    'title_li' => ''
));

Mess with the arguments as needed to get exactly what you want.