One of the things I like the most here is just to hang around the old city, there are plenty coffee houses and bars where you can take a moment to relax with a drink. On good weather days places usually are very full, but no need to worry, it is common just to stand in the middle of the street and the waiter will bring you the beer.
Visit the museum Kunstsammlung Nordrhein-Westfalen (Grabbeplatz 5). There are many beautiful paintings of the modern arts artists, like Dali, Kadinsky, Miró, Ernst, Lichtenstein, Mondrian and Picasso, which I like.
Souterrain Cinema is the nicest cinema I have ever seen. OK it's no big screen and no 'big' sound there but it's unique. To find it you have to go to Cafe Muggel in D-Oberkassel (Dominikanerstrasse 4) and downstairs you will find the cinema. It's basically a few chairs, tables, a bar and a screen. You can smoke during the film and there is a break because they have to change the film rolls. I love this place!