I activated the zenpage plugin, and the tinymice plugin. Took a little bit of thinkering, but it works. The only thing that doesn't work for me is how to make the page show up.
In the back end, I have the page. But I can't get it to show up on the site itself. Ony by going into the backend, open the page as admin and preview it.

What do I need to do?

You need to either use a theme with Zenpage plugin support or add that support to whatever theme used yourself. Info on how to do that on the theming tutorial.

I did. I used the default theme. I have modified it slightly (mainly CSS) to make it fit my site. But nothing more than that. My theme didn't have the news.php and pages.php. So I copied it from the default theme. But no go.

If you use the default theme as you say, you will have no access to the zenpage function.
You have to use the zenpage theme to display the news or made a theme to use the news.php and pages.php files.
You can also change the css of the zenpage theme in order to have graphics charts that suit your need.

You also can't copy one file from a theme to another. The Zenpage theme for example uses some custom functions which might cause errors.

The Efferscence+ Theme has also support for Zenpage as are there a few 3rd party themes. All listed on our site.