How To Customizing the Category Archive Page?

I’m using the Twenty Ten theme. I want to know which php files and how to update them so that I can put a large image in the list of posts on the category archive page. My page is this: https://www.rosemarysheel.com/category/blog. I put the posts format as gallery but all I get is a small thumbnail pic (and the text I want to get rid of “This gallery contains x# photos”). Again I want my category archive pages to have 1 large image with a small blurb of text with a link to the full post like on this site that I stumbles upon: https://anamericaninitaly.com/category/culture/

Please Advise!