how do I get the featured image on archive/category pages of wp

I am trying to make the featured image/thumbnail show up in the category/archive page on the left side of the content.

I believe the this code goes somewhere
<?php the_post_thumbnail(); ?>

but I have not found the right place yet and when I did get it to show up, it just came up for one post. I don't want to use a plugin if it can be fixed another way. (I have way too many and will need to cut some back already)

I am also trying to find where I can make the "read more" link at the end of the excerpt. I did find a plugin for this. At what point do you have too many plugins? (I know... loaded question) thank you!

here is a page example and screenshot attached.