Author and Date Information in Drupal 6
The confusing things about these settings in Drupal 6 is that they aren't in any of the content areas. In fact, it's a content setting that's controlled by the theme.
To turn off the author and date information, go to Administer > Site building > Themes > Global setting > Display post information on. That's not exactly intuitive.
Author and Date Information in Drupal 7
In Drupal 7 the author and date settings have moved. To find them in Drupal 7, you need to look inside the Content types. To get there, click on Structure and then Content types. Then click edit next to the Content type you want to turn off author and date information for.
Scroll down to the bottom of the page and click on the Display settings tab. Here you can uncheck the box and turn off the author and date information.
Unfortunately you can't get any more specific than this. For example, you can't turn one on but leave the other off. You also can't turn these on and off for specific content items. To do that, you'll need to edit the theme files.