There are many sites with information, guides, and GPS tracks available for you to plan a great run, ride or hike.
Both Komoot and Alltrails have a large selection of cycling tracks, perfect for short and long racing bike rides in the Aude countryside. Most are racing bike rides with a wide variety of distances and altitudes, but there are some everyday rides (into Carcassonne or to Castelnaudary along the Canal du Midi) as well. Also included are some hikes and runs.
Strava has racing bike and other tracks from different people who have been in the area as well. Search for “Le Trabet”, or simply do an activity search around Moussoulens or Le Trabet to find tracks.
UtagawaVTT has many Mountainbike (MTB / VTT) tracks. Search on their map for rides in the area.
Visorando has many good walking and hiking tracks, complete with descriptions, map and GPX tracks. Search around Moussoulens to find them.
Climbingaway has links to climbing sites, guides, etc. Just search the area around Carcassonne. For more climbing, search in the Pyrenees which aren’t far away (1,5h drive).