You could take a train from Paris/Gare de Lyon to Narbonne and then a local train to Carcassone from Narbonne. See schedules at www.Oui.sncf.
See other travel possibilities at www.Rome2Rio.com
PS. Basic train connections now are offered via Google maps. Type the departure city, the final arrival city and the symbol for train travel.
Voila the basic possibilities are depicted on the map including total estimated journey time!
There are a few direct trains from Lyon to Carcassonne, so another option (on a working week day would be best) could be to travel in the morning to Lyon, find a cheap restaurant in Lyon near the station for lunch, then get any train after lunch either direct to Carcassonne or via a change at Narbonne.
The food in Lyon is often very good, and so should IMO be worth the extra effort and travel time.
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