Visit Cathar Country/ Occitanie
The regions controlled by Raymond VI of Toulouse, Raymond Roger Trencavel, and the Comte of Foix (and lesser nobles and vassals) are now called Occitanie or sometimes the Occitan. When Pope Innocent III called for a crusade to eradicate the Cathar heretics, the invading army overran this region, first taking the walled cities in the open plains and later the castles in the foothills of the Pyrenees.
The mountain strongholds are sometimes called "Cathar castles" but that is a bit of a misnomer; the Cathars themselves did not have armies. The castles were defended by nobles who felt honor bound to shelter Cathars (in some cases their wives, sisters, daughters or other close family members).
Typically the Crusader army approached a city or castle and demanded that the Lord hand over the heretics; many of the nobles refused; and then the city or castle was besieged.
The walled city of Carcassonne is a must-see if you visit this area. However, its looks different now than in the 13th century. There are now three rings of curtain walls; in the 13th century, there were two. Extensive restoration made Carcassonne beautiful, but much of the work was not period-correct. Inside, it's full of cafes and gift shops. The museum housed in the Chateau Comtal is worth a visit. Carcassonne had not been taken for 800 years; the Crusaders took it by treachery - they held Raymond Roger Trencavel prisoner when he came outside the walls to parley.
Montsegur is famous as the place where defenders of the Cathars made a last stand. It is on a pog (steep hill):
This photo shows the steepest side; there are easier paths on other sides. The defenders only place light guards to watch this steep slope, assuming no one would attack that way. The crusaders hired professional Basque climbers to ascend at night, and after they gained a foothold on top of the pog, it was easy for them to take Montsegur. About 200 Cathars were burned.
Many other places are worth a visit. The photo below shows Minerve, a village in a rocky location surrounded by deep limestone valleys that initially prevented attackers from getting their trebuchets close to the town, but eventually the attackers found a foothold. There is a replica of a large trebuchet called "La mal voisin" (the bad neighbor") and a small museum. This town is refreshingly free of tourist oriented shops. People do live there.
Here is the castle at Puivert (Eleanor of Aquitaine may have visited it).
Even if you are not a fan of guided tours, you may want to consider one if you want visit remote castles. The mountain roads are winding, steep, and narrow in places; it's sometimes challenging to find the way. Parking is limited.
There seem to be more companies offering tours in this region now than in past. I had a very positive experience with James and Sophie at Cathar Country Tours. Their website is a great source of information, including photos.