Excluding a category from a post list

This is the basic scenario: I have 20 categories set up. On the main page, I have two columns at the bottom. One displays only posts in Category #1 — which was easy to do. My problem is with getting the posts from the other 19 categories to display in the second column. I’ve tried a variety of methods, and none of them have worked ideally — some only display the first category listed, some display only Category #1.

Any ideas?