create a menu which filters by categories

Hi! I need some newbie help with the category system. I'm using tagadelic and have created categories for my content, and would like to create a category menu, which when you select one of the categories, it filters the blog posts. I have managed to do this for the individual tags, but that is not what I need: I need to be able to place the names of the categories ( the actual categories such as general, education, etc.) in a menu, and not the individual tags. So, I want the link to filter the posts, and not to send me to the page with the tag cloud.
Another detail is that I would like to make it available for any user to tag a post at any time, instead of only the one who publishes it at the moment of its creation...

Thanks!