How to best add a WordPress Categories page?

[djw]

I am moving a site from HTML to WordPress and the previous site has many, many posts that need to be categorized. The old site simply listed these posts alphabetically, but I would like to have a page of categories that lists the name of the posts beneath the category title, rather than a long alphabetized list of posts, as before. This 'Categories' page needs to work like an index.

I can't find a plugin to do this, and I need it to work on a page, not just in a widget. I also don't want it to be like the archives and list every user, categories by month and that sort of thing. I also need new post titles added automatically to this list as the posts are published.

If anyone has seen or heard of something like this, please let me know I have tried 'List category posts', and 'Page-list', but they don't quite do what I am looking for.

Any help is appreciated. Thanks!