Site divided into two categories — pulling content from one to the other?

I have a blog with two categories, A and B.

I’d like to have a different display for A and for B — the A category has a different “look” than B. The blog will basically have a front page that allows you to choose whether you want to view the A section or the B section.

But I want, at the top of A, to have a box that says “The latest post in B:” and then, pulled from B, the TITLE of the latest post in B, and a 100x100px image from the latest post in B. If you click that box, you get moved over to blog category B.

Similarly, at the top of the “B” blog, I want to have “Recent news in A:”, then, pulled from A, the TITLE of the latest post in A, and a 100x100px image from the latest post in A. If you click that box, you get moved over to blog category A.

Where do I begin to look for information on how to do this? I’m stumbling along okay with CSS and some WP PHP codes, but the more sophisticated stuff is still beyond me. I’m even having trouble figuring out how to SEARCH for answers for this.