Part of the fantastic thing about the experience of Odense, I think, is the experience of the many different time periods. "Brandts" is a well-preserved textile mill from the late 1800s (1869 I think?), which today houses cafes, exhibitions, design shops and more. The area also contains Kulturmaskinen, which pulsates with activity, and in the same building You can visit Tidens Samling, a private museum, which shows Danish design, furnishings and clothing from the 1900s all the way up to the 1990s. At Brandts Klædefabrik you can definitely find art and culture for the whole family.
When you stand in the middle of the many beautiful urban spaces, remember to look up. Enjoy the architecture with factory windows, bricks and concrete. And if you have a little extra time, I would recommend that you go exploring for the beautiful gable paintings in Brandts Passage and in Farvergården. Especially the pink wings are good for an "Odense selfie".