How to start?

How can I start using the plugin?

Here is the documentation: http://directorist.com/

Is there any shortcode I should place on a page in order to view the listings at frontend?

The plugin automatically creates necessary pages with shortcodes at the time of activating it. Details can be found here http://directorist.com/docs/page-settings/required-pages/. You can control almost everything going Directory listings >> Directory Settings and also using shortcode parameters (http://directorist.com/docs/shortcodes/shortcodes/).

Is the themes included in the downloaded plugin? Do I need to install the themes separately?

No, the themes are not included in the downloaded plugin. You can continue using the theme you are using with the plugin, its not mandatory to use any specific theme with the plugin.