Having an image appear when a post is newly published.?

That should be possible by modifying the Loop in your theme’s index.php template file but how would you define “new”?

I’d like to have the post title, and then when I publish a new post a little image that says “New” would appear beside the title.

I would have to be able to give it a period of time that the image of “New” beside the post title, but I haven’t the faint idea what code I would need to do to make something like this work

one possibility: https://www.ads-software.com/support/topic/mark-new-posts-as-new?replies=4

details depend on your used theme.

try forum search https://www.ads-software.com/search/mark+new+post

I’ve previously used something like:

/* Add span to title on newly posted posts/pages Requires my_is_new() */
function my_highlight_new($title) {
	global $post, $my_options;
	if( !is_attachment() && in_the_loop() )  $title .= my_is_new();
	return $title;
}
add_action('the_title', 'my_highlight_new');

/*
Determine if a given post/page can be classed as 'new'. Used by  my_highlight_new() */
function my_is_new() {
	global $post, $my_options;
	$diff = date('U') - get_the_time('U', $post);
	if ( $diff < 604800 ) return ' <span class="new"></span>'; // default: 1 week
}

in a theme’s functions,php file and then used CSS to add an image on the class .new.