You do not want to edit any files. Instead create a new theme that is a sub theme of the theme you are using and create a new CSS file in there, with the new rules you need.

https://www.drupal.org/node/225125

The other option is a custom module adding the new CSS file on hook_init, but going the subtheme is generally the best route.

From your home page, right click on the panel that is spaced wrong and select "inspect element" (Chrome and Firefox). That will show you the DOM tree and the style rules applied to it. You may have to work up the tree, but find the one giving the padding and/or margins that is causing the problem. Copy that style rule to your new CSS sheet and adjust the padding and/or margin to what you want.

This is a pure CSS fix that is one of the basic fundamentals of building any site (Drupal or not). It won't affect the staff's ability to create announcements, as the fixes would be universal throughout the site. 
Jamie Holly
http://hollyit.net
On 11/28/2014 11:29 AM, Darrell Eifert wrote:
Hello Tony --

Not sure ... we've pretty much used themes and modules "out of the box" in creating the D6 site and they've worked remarkably well.   Our staff is trained to use the Panels UI and creates announcement 'snippets' on a regular basis to update the main page.  If I could keep this functionality that would be fine -- what kind of css modification were you thinking of, and to what files?

-- Darrell

On 11/28/2014 10:55 AM, Tony wrote:

Would it not be easier to do it with css?

On Nov 28, 2014 7:49 AM, "Darrell Eifert" <deifert@hampton.lib.nh.us> wrote:
Hello --

We just upgraded a copy of our Drupal 6 site using the Zen / CTI-Flex
theme to Drupal 7 using an AtaptiveTheme base with the Mix and Match
theme.  Everything went smoothly, except for how Panels behaves in the
new D7 theme.  We have a simple two-column center panel between two side
columns (see http://www.hampton.lib.nh.us for the current D6 production
site), and the D7 version of Panels creates an unacceptably huge amount
of vertical white space between the panels.  If I change each individual
panel to "no markup", the vertical panel space shrinks, but I also lose
the side and center margins of the two columns, which I would like to keep.

Before I start trying to change the Panels module code, does anyone have
a simple suggesting for reducing the vertical space between panels in a
column?

Thanks in advance ...

Darrell Eifert
Lane Memorial Library
--
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