Display posts in date-by-category order

I’d like to display posts in category order and then, within each category, in date order. As well, I want to format them differently depending on which category they belong to. (Each post will belong to only one category.)

I think I can manage the formatting part of it using the example in the codex, but I’m not sure about how to order the posts.

Basically, this site will have two categories: “topstory” and “normal”. A “topstory” is a breaking, important news item and “normal” is for ongoing news items. I want “topstories” to display at the top of the site and “normal” stories to follow — both categories presented in date order.

I’ve been reading about The Loop and I’m pretty sure this is where I should be concentrating, but it’s not clear to me how to do this.

Any suggestions?

Thanks.