Uhh… how to edit categories !?

I have searched everywhere, but I simply cannot find a category.php or similiar file for me to edit the content of the categories. I would like to just display the titles(perma-link) of the articles, and I don’t want to show the body under the category section.

Also, another thing I want to edit is how to, on the main page of the website, show just the article titles for each category, instead of having them organized by the time of the post. In another word, I want only the article topics to show up on the main page that is sorted by both category and time of post.

Thank you for your time !