How do I modify a Drupal Theme? The theme I am using has a slider -- a panel of automatically changing images to display a topic that the user can choose -- but it only shows a few slides.  I want to extend it to a dozen slides.  I did simple text searches to determine which files in the theme are used code files to provide the functionality to the slider.  That was the easy part.  I was able to extend the code from the few slides to 12 slides.

Now comes the more difficult and hidden part.  As I see it, each slide has two parts that are not hard coded.  First, each slide has a caption -- a text description.  Secondly, each slide has a URL associated with it that is where the browser goes if the user clicks on the slide.  These two things are not hard coded in any of the PHP or other files, as far as I can tell.  They can be edited by the admin in the drupal backend admin settings.

So I assumed that these fields are somehow set and retained in the database.  But I did a search on a downloaded copy of the database and I did not find this text.

Please advise.  What additional work do I need to do to complete this task?

I suppose I could just forge ahead.  Obviously this data is being stored somewhere.  Maybe they are stored in the database in an encrypted form for security reasons.  I could do an ftp upload of the site theme files.  But then what?  Do I uninstall the theme and then reinstall it?  Do I just deactivate it and then reactivate it?  How do I make my code changes work?
If I understand your (main) question correctly, you want to show additional images in the slider. In that case you have to narrow down which modules are used to create the slider. Typically a View is used to retrieve all content (stored in your database, normally not encrypted), an overview of all Views can be found at admin/structure/views. In the View there is an option to set the number of items that have to be shown in the Pager section (setting the number of items to show to 12 would in a typical installation be the thing you want to do).

When this is indeed the set up and you don't see the expected change after altering the value, don't forget to clear the caches (drush cr or via admin/config/development/performance).

If your environment is different, please provide additional details: which module is used to create the slider, e.g. Flex Slider, is Views used to provide the data (images provided by the user), etc.