Display html in posts on home page

… also, a “read more >>” link would be nice to add at the end of the excerpt

Hi Ken.

Shortcodes and HTML are stripped from the post excerpt. You can modify that functionality with a plugin or a function, both of which are discussed in this article:
https://marketblog.envato.com/web-design/wordpress-excerpts/

For a basic “read more” link, add this function to a child theme functions.php file:

function hu_excerpt_more( $more ) {
   return '<a href="'. get_permalink($post->ID) . '"> Read More >></a>';
}

If you wanted to get fancy and style it like a button, see this article:
https://www.elegantthemes.com/blog/tips-tricks/how-to-customize-the-wordpress-read-more-link-text