Hello Mihai,

You can achieve that in two ways:

1. Just add a "Read More" as the first thing in the article. Then configure the Joomla settings so it shows the things you need.
By using this approach you will need to edit all your articles in order to implement a "Read More".

2. You can hide the things that you do not want to be shown with CSS. Just inspect your website with Firebug or Chrome Inspect Element and then add your custom CSS code to your custom.scss file.
Here's an example code that you can use:

.blog {

	article {

		p {

			display: none;

		}

	}

}

I cannot guarantee that the above code will work with RT templates or others or if you have different markup (HTML) in your articles. But you just need to inspect your website with Firefox or Chrome Inspect Element and you can easily determine what CSS you should use.

Here, have a look at this video that I recorded for you (watch it in Fullscreen and FullHD for better quality). It shows everything that I explained above.