One of the simply must see or hard to miss even sights around Carcassonne is the Canal du Midi. It stretches over two hundred and forty kilometres in length bringing together the Garonne River and the Étang de Thau going from Toulouse.
Its really up to you how you choose to explore it. You can simply walk along at your own pace taking it all in or you could rent a bike and cycle to Trèbes or Castelnaudary.
Another option worth looking into is the possibility of seeing the route by boat.
There are trips run by guides or you may choose to let your imagination tell the story.
The work of Pierre-Paul Riquet here has been described as a monumental feat in engineering.