Let user subscribe to specific categories

You can give them this url http://www.example.com/?cat=42&feed=rss2

This one if permalinks are enabled http://example.com/category/categoryname/feed

Replace the site url & also the cateogries id or slug with the correct one

UPDATE IN RESPONSE TO COMMENT

Display a list of categories to the user maybe with wp_dropdown_categories. Save the id of the category the user subscribed to in the database by using the function add_user_meta() or update_user_meta, then on the template on which you want to show the posts use WP_Query to retrieve all the posts.

If you don’t know the functions, check out their reference on codex, it’s easy to find on google