WordPress should automatically be displaying archives for each category using files from your theme (or child theme – more on that lower down) from the file hierarchy.
The specific order is:
- category-slug.php
- category-ID.php
- category.php
- archive.php
- index.php
It sounds like you need to create or edit your category.php file unless you have ID or slug based customisations. Check the documentation if in doubt.
As for your 404 errors, check the permalinks section in your admin area. Sometimes (especially after making changes) it needs you to hit save there to flush through the changes.
Further reading
Customising archives
This guide goes into detail about customising archives but it is just one of many tutorials on the web.
Child themes
A child theme is a way to make a theme based on another theme without updates to the parent undoing all your work. I assume that this is what you have done in customising your theme. If not Google the subject because there is a lot of good content already written about creating a child theme.